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Chapter 7Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860184-3 GeometryChapter 7 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 7-1Study Guide and Intervention . . . . . . . . 351–352Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 353Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Reading to Learn Mathematics . . . . . . . . . . 355Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 356
Lesson 7-2Study Guide and Intervention . . . . . . . . 357–358Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 359Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 360Reading to Learn Mathematics . . . . . . . . . . 361Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 362
Lesson 7-3Study Guide and Intervention . . . . . . . . 363–364Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 365Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 366Reading to Learn Mathematics . . . . . . . . . . 367Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 368
Lesson 7-4Study Guide and Intervention . . . . . . . . 369–370Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 371Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 372Reading to Learn Mathematics . . . . . . . . . . 373Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 374
Lesson 7-5Study Guide and Intervention . . . . . . . . 375–376Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 377Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 378Reading to Learn Mathematics . . . . . . . . . . 379Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 380
Lesson 7-6Study Guide and Intervention . . . . . . . . 381–382Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 383Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 384Reading to Learn Mathematics . . . . . . . . . . 385Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 386
Lesson 7-7Study Guide and Intervention . . . . . . . . 387–388Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 389Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 390Reading to Learn Mathematics . . . . . . . . . . 391Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 392
Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 393–394Chapter 7 Test, Form 2A . . . . . . . . . . . 395–396Chapter 7 Test, Form 2B . . . . . . . . . . . 397–398Chapter 7 Test, Form 2C . . . . . . . . . . . 399–400Chapter 7 Test, Form 2D . . . . . . . . . . . 401–402Chapter 7 Test, Form 3 . . . . . . . . . . . . 403–404Chapter 7 Open-Ended Assessment . . . . . . 405Chapter 7 Vocabulary Test/Review . . . . . . . 406Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 407Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 408Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 409Chapter 7 Cumulative Review . . . . . . . . . . . 410Chapter 7 Standardized Test Practice . 411–412Unit 2 Test/Review (Ch. 4–7) . . . . . . . . 413–414First Semester Test (Ch. 1–7) . . . . . . . 415–416
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A34
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 7 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 7Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 398–399. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
ambiguous case
angle of depression
angle of elevation
cosine
geometric mean
Law of Cosines
Law of Sines
Pythagorean identity
puh·thag·uh·REE·ahn
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
Pythagorean triple
reciprocal identity
ri·SIP·ruh·kuhl
sine
solve a triangle
tangent
trigonometric identity
trig·uh·nuh·MET·rik
trigonometric ratio
trigonometry
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
77
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 7. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 7.1
Theorem 7.2
Theorem 7.3
Theorem 7.4Pythagorean Theorem
Theorem 7.5Converse of the Pythagorean Theorem
Theorem 7.6
Theorem 7.7
Study Guide and InterventionGeometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 351 Glencoe Geometry
Less
on
7-1
Geometric Mean The geometric mean between two numbers is the square root oftheir product. For two positive numbers a and b, the geometric mean of a and b is the positive number x in the proportion �
ax� � �b
x�. Cross multiplying gives x2 � ab, so x � �ab�.
Find the geometric mean between each pair of numbers.
a. 12 and 3Let x represent the geometric mean.
�1x2� � �3
x� Definition of geometric mean
x2 � 36 Cross multiply.
x � �36� or 6 Take the square root of each side.
b. 8 and 4Let x represent the geometric mean.
�8x� � �4
x�
x2 � 32x � �32�
� 5.7
ExercisesExercises
Find the geometric mean between each pair of numbers.
1. 4 and 4 2. 4 and 6
3. 6 and 9 4. �12� and 2
5. 2�3� and 3�3� 6. 4 and 25
7. �3� and �6� 8. 10 and 100
9. �12� and �
14� 10. and
11. 4 and 16 12. 3 and 24
The geometric mean and one extreme are given. Find the other extreme.
13. �24� is the geometric mean between a and b. Find b if a � 2.
14. �12� is the geometric mean between a and b. Find b if a � 3.
Determine whether each statement is always, sometimes, or never true.
15. The geometric mean of two positive numbers is greater than the average of the twonumbers.
16. If the geometric mean of two positive numbers is less than 1, then both of the numbersare less than 1.
3�2��5
2�2��5
ExampleExample
© Glencoe/McGraw-Hill 352 Glencoe Geometry
Altitude of a Triangle In the diagram, �ABC � �ADB � �BDC.An altitude to the hypotenuse of a right triangle forms two right triangles. The two triangles are similar and each is similar to the original triangle. CD
B
A
Study Guide and Intervention (continued)
Geometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Use right �ABC with B�D� ⊥ A�C�. Describe two geometricmeans.
a. �ADB � �BDC so �BAD
D� � �BC
DD�.
In �ABC, the altitude is the geometricmean between the two segments of thehypotenuse.
b. �ABC � �ADB and �ABC � �BDC,
so �AACB� � �A
ADB� and �B
ACC� � �D
BCC�.
In �ABC, each leg is the geometricmean between the hypotenuse and thesegment of the hypotenuse adjacent tothat leg.
Find x, y, and z.
�PP
QR� � �
PP
QS�
�21
55� � �
1x5� PR � 25, PQ � 15, PS � x
25x � 225 Cross multiply.
x � 9 Divide each side by 25.
Theny � PR � SP
� 25 � 9� 16
�QPR
R� � �QR
RS�
�2z5� � �y
z� PR � 25, QR � z, RS � y
�2z5� � �1
z6� y � 16
z2 � 400 Cross multiply.
z � 20 Take the square root of each side.
z
y
x
15
R
Q P
S25
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x, y, and z to the nearest tenth.
1. 2. 3.
4. 5. 6.x zy
62
x
z y
2
2xy
1
��3
��12
zxy
81
z
xy 5
2
x
1 3
Skills PracticeGeometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 353 Glencoe Geometry
Less
on
7-1
Find the geometric mean between each pair of numbers. State exact answers andanswers to the nearest tenth.
1. 2 and 8 2. 9 and 36 3. 4 and 7
4. 5 and 10 5. 2�2� and 5�2� 6. 3�5� and 5�5�
Find the measure of each altitude. State exact answers and answers to the nearesttenth.
7. 8.
9. 10.
Find x and y.
11. 12.
13. 14.
2
5y
x
15
4
y
x
10
4
yx
3 9
yx
R T
S
U4.5 8G
E H
F
2
9
L
M
N
P 2
12
C
D
B
A 2
7
© Glencoe/McGraw-Hill 354 Glencoe Geometry
Find the geometric mean between each pair of numbers to the nearest tenth.
1. 8 and 12 2. 3�7� and 6�7� 3. �45� and 2
Find the measure of each altitude. State exact answers and answers to the nearesttenth.
4. 5.
Find x, y, and z.
6. 7.
8. 9.
10. CIVIL ENGINEERING An airport, a factory, and a shopping center are at the vertices of aright triangle formed by three highways. The airport and factory are 6.0 miles apart. Theirdistances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service roadwill be constructed from the shopping center to the highway that connects the airport andfactory. What is the shortest possible length for the service road? Round to the nearesthundredth.
x y
10z
20x
y
2
3
z
zx y
625
23
z
xy
8
17
6
KL
J M
125
U
T A V
Practice Geometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Reading to Learn MathematicsGeometric Mean
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 355 Glencoe Geometry
Less
on
7-1
Pre-Activity How can the geometric mean be used to view paintings?
Read the introduction to Lesson 7-1 at the top of page 342 in your textbook.
• What is a disadvantage of standing too close to a painting?
• What is a disadvantage of standing too far from a painting?
Reading the Lesson1. In the past, when you have seen the word mean in mathematics, it referred to the
average or arithmetic mean of the two numbers.
a. Complete the following by writing an algebraic expression in each blank.
If a and b are two positive numbers, then the geometric mean between a and b is
and their arithmetic mean is .
b. Explain in words, without using any mathematical symbols, the difference betweenthe geometric mean and the algebraic mean.
2. Let r and s be two positive numbers. In which of the following equations is z equal to thegeometric mean between r and s?
A. �zs
� � �zr� B. �z
r� � �z
s� C. s : z � z: r D. �z
r� � �
zs� E. �
zr� � �
zs� F. �
zs� � �z
r�
3. Supply the missing words or phrases to complete the statement of each theorem.
a. The measure of the altitude drawn from the vertex of the right angle of a right triangle
to its hypotenuse is the between the measures of the two
segments of the .
b. If the altitude is drawn from the vertex of the angle of a right
triangle to its hypotenuse, then the measure of a of the triangle
is the between the measure of the hypotenuse and the segment
of the adjacent to that leg.
c. If the altitude is drawn from the of the right angle of a right
triangle to its , then the two triangles formed are
to the given triangle and to each other.
Helping You Remember4. A good way to remember a new mathematical concept is to relate it to something you
already know. How can the meaning of mean in a proportion help you to remember howto find the geometric mean between two numbers?
© Glencoe/McGraw-Hill 356 Glencoe Geometry
Mathematics and MusicPythagoras, a Greek philosopher who lived during the sixth century B.C.,believed that all nature, beauty, and harmony could be expressed by whole-number relationships. Most people remember Pythagoras for his teachingsabout right triangles. (The sum of the squares of the legs equals the square ofthe hypotenuse.) But Pythagoras also discovered relationships between themusical notes of a scale. These relationships can be expressed as ratios.
C D E F G A B C�
�11� �
89� �
45� �
34� �
23� �
35� �1
85� �
12�
When you play a stringed instrument, The C string can be usedyou produce different notes by placing to produce F by placingyour finger on different places on a string. a finger �
34� of the way
This is the result of changing the lengthalong the string.of the vibrating part of the string.
Suppose a C string has a length of 16 inches. Write and solve proportions to determine what length of string would have to vibrate to produce the remaining notes of the scale.
1. D 2. E 3. F
4. G 5. A 6. B
7. C�
8. Complete to show the distance between finger positions on the 16-inch
C string for each note. For example, C(16) � D�14�29�� � 1�
79�.
C D E F G A B C�
9. Between two consecutive musical notes, there is either a whole step or a half step. Using the distances you found in Exercise 8, determine what two pairs of notes have a half step between them.
1�79� in.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
34 of C string
Study Guide and InterventionThe Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 357 Glencoe Geometry
Less
on
7-2
The Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.
�ABC is a right triangle, so a2 � b2 � c2.
Prove the Pythagorean Theorem.With altitude C�D�, each leg a and b is a geometric mean between hypotenuse c and the segment of the hypotenuse adjacent to that leg.
�ac
� � �ay� and �b
c� � �
bx�, so a2 � cy and b2 � cx.
Add the two equations and substitute c � y � x to geta2 � b2 � cy � cx � c( y � x) � c2.
c y
x a
b
hA C
BD
c a
bA C
B
Example 1Example 1
Example 2Example 2
a. Find a.
a2 � b2 � c2 Pythagorean Theorem
a2 � 122 � 132 b � 12, c � 13
a2 � 144 � 169 Simplify.a2 � 25 Subtract.
a � 5 Take the square root of each side.
a
12
13
AC
B
b. Find c.
a2 � b2 � c2 Pythagorean Theorem
202 � 302 � c2 a � 20, b � 30
400 � 900 � c2 Simplify.
1300 � c2 Add.
�1300� � c Take the square root of each side.
36.1 � c Use a calculator.
c
30
20
AC
B
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6.x
1128
x
33
16x
59
49
x
6525
x
159
x
3 3
© Glencoe/McGraw-Hill 358 Glencoe Geometry
Converse of the Pythagorean Theorem If the sum of the squares of the measures of the two shorter sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.
If the three whole numbers a, b, and c satisfy the equation a2 � b2 � c2, then the numbers a, b, and c form a If a2 � b2 � c2, then
Pythagorean triple. �ABC is a right triangle.
Determine whether �PQR is a right triangle.a2 � b2 � c2 Pythagorean Theorem
102 � (10�3�)2 � 202 a � 10, b � 10�3�, c � 20
100 � 300 � 400 Simplify.
400 � 400✓ Add.
The sum of the squares of the two shorter sides equals the square of the longest side, so thetriangle is a right triangle.
Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.
1. 30, 40, 50 2. 20, 30, 40 3. 18, 24, 30
4. 6, 8, 9 5. �37�, �
47�, �
57� 6. 10, 15, 20
7. �5�, �12�, �13� 8. 2, �8�, �12� 9. 9, 40, 41
A family of Pythagorean triples consists of multiples of known triples. For eachPythagorean triple, find two triples in the same family.
10. 3, 4, 5 11. 5, 12, 13 12. 7, 24, 25
10��3
20 10
QR
P
c
ab
A
C
B
Study Guide and Intervention (continued)
The Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
ExampleExample
ExercisesExercises
Skills PracticeThe Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 359 Glencoe Geometry
Less
on
7-2
Find x.
1. 2. 3.
4. 5. 6.
Determine whether �STU is a right triangle for the given vertices. Explain.
7. S(5, 5), T(7, 3), U(3, 2) 8. S(3, 3), T(5, 5), U(6, 0)
9. S(4, 6), T(9, 1), U(1, 3) 10. S(0, 3), T(�2, 5), U(4, 7)
11. S(�3, 2), T(2, 7), U(�1, 1) 12. S(2, �1), T(5, 4), U(6, �3)
Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.
13. 12, 16, 20 14. 16, 30, 32 15. 14, 48, 50
16. �25�, �
45�, �
65� 17. 2�6�, 5, 7 18. 2�2�, 2�7�, 6
x
31
14x9 9
8
x12.5
25
x
1232
x
12
13x
12
9
© Glencoe/McGraw-Hill 360 Glencoe Geometry
Find x.
1. 2. 3.
4. 5. 6.
Determine whether �GHI is a right triangle for the given vertices. Explain.
7. G(2, 7), H(3, 6), I(�4, �1) 8. G(�6, 2), H(1, 12), I(�2, 1)
9. G(�2, 1), H(3, �1), I(�4, �4) 10. G(�2, 4), H(4, 1), I(�1, �9)
Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.
11. 9, 40, 41 12. 7, 28, 29 13. 24, 32, 40
14. �95�, �
152�, 3 15. �
13�, , 1 16. , , �
47�
17. CONSTRUCTION The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long. How high is the dock?
11 ft?
dock
ramp
10 ft
2�3��7
�4��7
2�2��3
x2424
42
x16
14
x
34
22
x26
2618
x
34 21x
13
23
Practice The Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
Reading to Learn MathematicsThe Pythagorean Theorem and Its Converse
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 361 Glencoe Geometry
Less
on
7-2
Pre-Activity How are right triangles used to build suspension bridges?
Read the introduction to Lesson 7-2 at the top of page 350 in your textbook.
Do the two right triangles shown in the drawing appear to be similar?Explain your reasoning.
Reading the Lesson
1. Explain in your own words the difference between how the Pythagorean Theorem is usedand how the Converse of the Pythagorean Theorem is used.
2. Refer to the figure. For this figure, which statements are true?
A. m2 � n2 � p2 B. n2 � m2 � p2
C. m2 � n2 � p2 D. m2 � p2 � n2
E. p2 � n2 � m2 F. n2 � p2 � m2
G. n � �m2 ��p2� H. p � �m2 ��n2�
3. Is the following statement true or false?A Pythagorean triple is any group of three numbers for which the sum of the squares of thesmaller two numbers is equal to the square of the largest number. Explain your reasoning.
4. If x, y, and z form a Pythagorean triple and k is a positive integer, which of the followinggroups of numbers are also Pythagorean triples?
A. 3x, 4y, 5z B. 3x, 3y, 3z C. �3x, �3y, �3z D. kx, ky, kz
Helping You Remember
5. Many students who studied geometry long ago remember the Pythagorean Theorem as theequation a2 � b2 � c2, but cannot tell you what this equation means. A formula is uselessif you don’t know what it means and how to use it. How could you help someone who hasforgotten the Pythagorean Theorem remember the meaning of the equation a2 � b2 � c2?
pm
n
© Glencoe/McGraw-Hill 362 Glencoe Geometry
Converse of a Right Triangle TheoremYou have learned that the measure of the altitude from the vertex ofthe right angle of a right triangle to its hypotenuse is the geometricmean between the measures of the two segments of the hypotenuse.Is the converse of this theorem true? In order to find out, it will helpto rewrite the original theorem in if-then form as follows.
If �ABQ is a right triangle with right angle at Q, then QP is the geometric mean between AP and PB, where Pis between A and B and Q�P� is perpendicular to A�B�.
1. Write the converse of the if-then form of the theorem.
2. Is the converse of the original theorem true? Refer to the figure at the right to explain your answer.
You may find it interesting to examine the other theorems inChapter 7 to see whether their converses are true or false. You willneed to restate the theorems carefully in order to write theirconverses.
Q
BP
A
Q
BPA
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
Study Guide and InterventionSpecial Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 363 Glencoe Geometry
Less
on
7-3
Properties of 45°-45°-90° Triangles The sides of a 45°-45°-90° right triangle have aspecial relationship.
If the leg of a 45°-45°-90°right triangle is x units, show that the hypotenuse is x�2� units.
Using the Pythagorean Theorem with a � b � x, then
c2 � a2 � b2
� x2 � x2
� 2x2
c � �2x2�� x�2�
x��
x
x 245�
45�
In a 45°-45°-90° right triangle the hypotenuse is �2� times the leg. If the hypotenuse is 6 units,find the length of each leg.The hypotenuse is �2� times the leg, sodivide the length of the hypotenuse by �2�.
a �
�
�
� 3�2� units
6�2��2
6�2���2��2�
6��2�
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6.
7. Find the perimeter of a square with diagonal 12 centimeters.
8. Find the diagonal of a square with perimeter 20 inches.
9. Find the diagonal of a square with perimeter 28 meters.
x 3��2x 18x x
18
x10x
45�
3��2x
8
45�
45�
© Glencoe/McGraw-Hill 364 Glencoe Geometry
Properties of 30°-60°-90° Triangles The sides of a 30°-60°-90° right triangle alsohave a special relationship.
In a 30°-60°-90° right triangle, show that the hypotenuse is twice the shorter leg and the longer leg is �3� times the shorter leg.
�MNQ is a 30°-60°-90° right triangle, and the length of the hypotenuse M�N� is two times the length of the shorter side N�Q�.Using the Pythagorean Theorem,a2 � (2x) 2 � x2
� 4x2 � x2
� 3x2
a � �3x2�� x�3�
In a 30°-60°-90° right triangle, the hypotenuse is 5 centimeters.Find the lengths of the other two sides of the triangle.If the hypotenuse of a 30°-60°-90° right triangle is 5 centimeters, then the length of theshorter leg is half of 5 or 2.5 centimeters. The length of the longer leg is �3� times the length of the shorter leg, or (2.5)(�3�) centimeters.
Find x and y.
1. 2. 3.
4. 5. 6.
7. The perimeter of an equilateral triangle is 32 centimeters. Find the length of an altitudeof the triangle to the nearest tenth of a centimeter.
8. An altitude of an equilateral triangle is 8.3 meters. Find the perimeter of the triangle tothe nearest tenth of a meter.
xy
60�
20
xy60�
12
xy
30�
9��3
x
y
11
30�
x
y
60�
8
x
y30�
60�12
x
a
N
Q
P
M
2x
30�30�
60�
60�
Study Guide and Intervention (continued)
Special Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
ExercisesExercises
Example 1Example 1
Example 2Example 2
�MNP is an equilateraltriangle.
�MNQ is a 30°-60°-90°right triangle.
Skills PracticeSpecial Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 365 Glencoe Geometry
Less
on
7-3
Find x and y.
1. 2. 3.
4. 5. 6.
For Exercises 7–9, use the figure at the right.
7. If a � 11, find b and c.
8. If b � 15, find a and c.
9. If c � 9, find a and b.
For Exercises 10 and 11, use the figure at the right.
10. The perimeter of the square is 30 inches. Find the length of B�C�.
11. Find the length of the diagonal B�D�.
12. The perimeter of the equilateral triangle is 60 meters. Find the length of an altitude.
13. �GEC is a 30°-60°-90° triangle with right angle at E, and E�C� is the longer leg. Find the coordinates of G in Quadrant I for E(1, 1) and C(4, 1).
E
FGD 60�
A B
CD 45�
bA
B
C
ac
60�
30�
y
x�
13
1313
13
y
x60�
16
y
x
45� 8
y
x
45�
12
y
x
30�
32
y
x60� 24
© Glencoe/McGraw-Hill 366 Glencoe Geometry
Find x and y.
1. 2. 3.
4. 5. 6.
For Exercises 7–9, use the figure at the right.
7. If a � 4�3�, find b and c.
8. If x � 3�3�, find a and CD.
9. If a � 4, find CD, b, and y.
10. The perimeter of an equilateral triangle is 39 centimeters. Find the length of an altitudeof the triangle.
11. �MIP is a 30°-60°-90° triangle with right angle at I, and I�P� the longer leg. Find thecoordinates of M in Quadrant I for I(3, 3) and P(12, 3).
12. �TJK is a 45°-45°-90° triangle with right angle at J. Find the coordinates of T inQuadrant II for J(�2, �3) and K(3, �3).
13. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from adiagonal pathway through the garden. How long is the pathway?
6 yd 6 yd
6 yd
6 yd
bA
B
C
D
a
x
y60�
30�
c
x
45�
11
y60�3.5
xy
x�
y 28
y
x
30�
26y
x
25
60�
yx
45�9
Practice Special Right Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Reading to Learn MathematicsSpecial Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 367 Glencoe Geometry
Less
on
7-3
Pre-Activity How is triangle tiling used in wallpaper design?
Read the introduction to Lesson 7-3 at the top of page 357 in your textbook.• How can you most completely describe the larger triangle and the two
smaller triangles in tile 15?
• How can you most completely describe the larger triangle and the twosmaller triangles in tile 16? (Include angle measures in describing all thetriangles.)
Reading the Lesson1. Supply the correct number or numbers to complete each statement.
a. In a 45°-45°-90° triangle, to find the length of the hypotenuse, multiply the length of a
leg by .
b. In a 30°-60°-90° triangle, to find the length of the hypotenuse, multiply the length of
the shorter leg by .
c. In a 30°-60°-90° triangle, the longer leg is opposite the angle with a measure of .
d. In a 30°-60°-90° triangle, to find the length of the longer leg, multiply the length of
the shorter leg by .
e. In an isosceles right triangle, each leg is opposite an angle with a measure of .
f. In a 30°-60°-90° triangle, to find the length of the shorter leg, divide the length of the
longer leg by .
g. In 30°-60°-90° triangle, to find the length of the longer leg, divide the length of the
hypotenuse by and multiply the result by .
h. To find the length of a side of a square, divide the length of the diagonal by .
2. Indicate whether each statement is always, sometimes, or never true.a. The lengths of the three sides of an isosceles triangle satisfy the Pythagorean
Theorem.b. The lengths of the sides of a 30°-60°-90° triangle form a Pythagorean triple.c. The lengths of all three sides of a 30°-60°-90° triangle are positive integers.
Helping You Remember3. Some students find it easier to remember mathematical concepts in terms of specific
numbers rather than variables. How can you use specific numbers to help you rememberthe relationship between the lengths of the three sides in a 30°-60°-90° triangle?
© Glencoe/McGraw-Hill 368 Glencoe Geometry
Constructing Values of Square RootsThe diagram at the right shows a right isosceles triangle with two legs of length 1 inch. By the Pythagorean Theorem, the length of the hypotenuse is �2� inches. By constructing an adjacent right triangle with legs of �2� inches and 1 inch, you can create a segment of length �3�.
By continuing this process as shown below, you can construct a “wheel” of square roots. This wheel is called the “Wheel of Theodorus”after a Greek philosopher who lived about 400 B.C.
Continue constructing the wheel until you make a segment oflength �18�.
��
1
1
1
3��
2
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
1
1
11
1
1
��2
��3
��5
��6
��7
��8
��10
��11 ��12��13
��14
��15
��17
��18
��16 � 4
��4 � 2
��9 � 3
Study Guide and InterventionTrigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 369 Glencoe Geometry
Less
on
7-4
Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan,respectively.
sin R � �leg
hyopppotoesnitues�
eR
� cos R � tan R �
� �rt� � �
st� � �
rs�
Find sin A, cos A, and tan A. Express each ratio as a decimal to the nearest thousandth.
sin A � �ohpyppoostietneulseeg
� cos A � �ahdyjpaocteenntulseeg
� tan A � �aopd
pja
ocseintet
lleegg�
� �BAB
C� � �A
ABC� � �
BAC
C�
� �153� � �
11
23� � �1
52�
� 0.385 � 0.923 � 0.417
Find the indicated trigonometric ratio as a fraction and as a decimal. If necessary, round to the nearest ten-thousandth.
1. sin A 2. tan B
3. cos A 4. cos B
5. sin D 6. tan E
7. cos E 8. cos D
16
1620
12
3430
C
B
A D F
E
12
135
C
B
A
leg opposite �R���leg adjacent to �R
leg adjacent to �R���hypotenuse
s
tr
T
S
R
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 370 Glencoe Geometry
Use Trigonometric Ratios In a right triangle, if you know the measures of two sidesor if you know the measures of one side and an acute angle, then you can use trigonometricratios to find the measures of the missing sides or angles of the triangle.
Find x, y, and z. Round each measure to the nearest whole number. 1858�
x � CB y
zA
Study Guide and Intervention (continued)
Trigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
a. Find x.
x � 58 � 90x � 32
b. Find y.
tan A � �1y8�
tan 58° � �1y8�
y � 18 tan 58°y � 29
c. Find z.
cos A � �1z8�
cos 58° � �1z8�
z cos 58° � 18
z � �cos18
58°�
z � 34
ExercisesExercises
Find x. Round to the nearest tenth.
1. 2.
3. 4.
5. 6.15
64� x16
40�
x
4
1x�12
5x�
12 16
x�3228�
x
ExampleExample
Skills PracticeTrigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 371 Glencoe Geometry
Less
on
7-4
Use �RST to find sin R, cos R, tan R, sin S, cos S, and tan S.Express each ratio as a fraction and as a decimal to the nearest hundredth.
1. r � 16, s � 30, t � 34 2. r � 10, s � 24, t � 26
Use a calculator to find each value. Round to the nearest ten-thousandth.
3. sin 5 4. tan 23 5. cos 61
6. sin 75.8 7. tan 17.3 8. cos 52.9
Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to thenearest ten-thousandth.
9. tan C 10. sin A 11. cos C
Find the measure of each acute angle to the nearest tenth of a degree.
12. sin B � 0.2985 13. tan A � 0.4168 14. cos R � 0.8443
15. tan C � 0.3894 16. cos B � 0.7329 17. sin A � 0.1176
Find x. Round to the nearest tenth.
18. 19. 20.
19
x
33� UL
S
27
x �
8
BA
C
27
x �
13
BA
C
41
409B
A
C
sR
S
T
rt
© Glencoe/McGraw-Hill 372 Glencoe Geometry
Use �LMN to find sin L, cos L, tan L, sin M, cos M, and tan M.Express each ratio as a fraction and as a decimal to the nearest hundredth.
1. � � 15, m � 36, n� 39 2. � � 12, m � 12�3�, n � 24
Use a calculator to find each value. Round to the nearest ten-thousandth.
3. sin 92.4 4. tan 27.5 5. cos 64.8
Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to the nearest ten-thousandth.
6. cos A 7. tan B 8. sin A
Find the measure of each acute angle to the nearest tenth of a degree.
9. sin B � 0.7823 10. tan A � 0.2356 11. cos R � 0.6401
Find x. Round to the nearest tenth.
12. 13. 14.
15. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a vertical rockformation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36 degrees. What is the height of the formation to the nearest meter?
36�
43 m
41�x
3229
x �9
23
x �
11
15
5��105
CA
B
ML
N
Practice Trigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
Reading to Learn MathematicsTrigonometry
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 373 Glencoe Geometry
Less
on
7-4
Pre-Activity How can surveyors determine angle measures?
Read the introduction to Lesson 7-4 at the top of page 364 in your textbook.
• Why is it important to determine the relative positions accurately innavigation? (Give two possible reasons.)
• What does calibrated mean?
Reading the Lesson
1. Refer to the figure. Write a ratio using the side lengths in the figure to represent each of the following trigonometric ratios.
A. sin N B. cos N
C. tan N D. tan M
E. sin M F. cos M
2. Assume that you enter each of the expressions in the list on the left into your calculator.Match each of these expressions with a description from the list on the right to tell whatyou are finding when you enter this expression.
P
M N
a. sin 20
b. cos 20
c. sin�1 0.8
d. tan�1 0.8
e. tan 20
f. cos�1 0.8
i. the degree measure of an acute angle whose cosine is 0.8
ii. the ratio of the length of the leg adjacent to the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle
iii.the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe adjacent leg is 0.8
iv. the ratio of the length of the leg opposite the 20° angle to thelength of the leg adjacent to it in a 20°-70°-90° triangle
v. the ratio of the length of the leg opposite the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle
vi. the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe hypotenuse is 0.8
Helping You Remember
3. How can the co in cosine help you to remember the relationship between the sines andcosines of the two acute angles of a right triangle?
© Glencoe/McGraw-Hill 374 Glencoe Geometry
Sine and Cosine of AnglesThe following diagram can be used to obtain approximate values for the sineand cosine of angles from 0° to 90°. The radius of the circle is 1. So, the sineand cosine values can be read directly from the vertical and horizontal axes.
Find approximate values for sin 40°and cos 40�. Consider the triangle formed by the segment marked 40°, as illustrated by the shaded triangle at right.
sin 40° � �ac� � �
0.164� or 0.64 cos 40° � �
bc� � �
0.177� or 0.77
1. Use the diagram above to complete the chart of values.
2. Compare the sine and cosine of two complementary angles (angles whose sum is 90°). What do you notice?
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
90°
0°
10°
20°
30°
40°
50°
60°
70°80°
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
x° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90°
sin x° 0.64
cos x° 0.77
1
0
40°0.64
c � 1 unit
x °b � cos x ° 0.77 1
a � sin x °
ExampleExample
Study Guide and InterventionAngles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 375 Glencoe Geometry
Less
on
7-5
Angles of Elevation Many real-world problems that involve looking up to an object can be described in terms of an angle of elevation, which is the angle between an observer’s line of sight and a horizontal line.
The angle of elevation from point A to the top of a cliff is 34°. If point A is 1000 feet from the base of the cliff,how high is the cliff?Let x � the height of the cliff.
tan 34° � �10x00� tan � �
oapdpjaocseitnet
�
1000(tan 34°) � x Multiply each side by 1000.
674.5 � x Use a calculator.
The height of the cliff is about 674.5 feet.
Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.
1. The angle of elevation from point A to the top of a hill is 49°.If point A is 400 feet from the base of the hill, how high is the hill?
2. Find the angle of elevation of the sun when a 12.5-meter-tall telephone pole casts a 18-meter-long shadow.
3. A ladder leaning against a building makes an angle of 78°with the ground. The foot of the ladder is 5 feet from the building. How long is the ladder?
4. A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower.That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation?
?5 ft
100 ft
132 ft
78�5 ft
?
18 m
12.5 m
sun
?
✹
400 ft
?
49�A
?
1000 ft34�A
x
angle ofelevation
line of si
ght
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 376 Glencoe Geometry
Angles of Depression When an observer is looking down, the angle of depression is the angle between the observer’s line of sight and a horizontal line.
The angle of depression from the top of an 80-foot building to point A on the ground is 42°. How far is the foot of the building from point A?Let x � the distance from point A to the foot of the building. Since the horizontal line is parallel to the ground, the angle of depression�DBA is congruent to �BAC.
tan 42° � �8x0� tan � �
o
a
p
d
p
ja
o
c
s
e
it
n
e
t�
x(tan 42°) � 80 Multiply each side by x.
x � �tan80
42°� Divide each side by tan 42°.
x � 88.8 Use a calculator.
Point A is about 89 feet from the base of the building.
Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.
1. The angle of depression from the top of a sheer cliff to point A on the ground is 35°. If point A is 280 feet from the base of the cliff, how tall is the cliff?
2. The angle of depression from a balloon on a 75-foot string to a person on the ground is 36°. How high is the balloon?
3. A ski run is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the ski run to the bottom.
4. From the top of a 120-foot-high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19°. How far from the base of thetower is the airplane?
120 ft
?
19�
208 yd
?
1000 yd
36�
75 ft ?
A
35�
280 ft
?
A C
BD
x42�
angle ofdepression
horizontal
80 ft
Yline of sight
horizontalangle ofdepression
Study Guide and Intervention (continued)
Angles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
ExercisesExercises
ExampleExample
Skills PracticeAngles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 377 Glencoe Geometry
Less
on
7-5
Name the angle of depression or angle of elevation in each figure.
1. 2.
3. 4.
5. MOUNTAIN BIKING On a mountain bike trip along the Gemini Bridges Trail in Moab,Utah, Nabuko stopped on the canyon floor to get a good view of the twin sandstonebridges. Nabuko is standing about 60 meters from the base of the canyon cliff, and thenatural arch bridges are about 100 meters up the canyon wall. If her line of sight is fivefeet above the ground, what is the angle of elevation to the top of the bridges? Round tothe nearest tenth degree.
6. SHADOWS Suppose the sun casts a shadow off a 35-foot building.If the angle of elevation to the sun is 60°, how long is the shadow to the nearest tenth of a foot?
7. BALLOONING From her position in a hot-air balloon, Angie can see her car parked in afield. If the angle of depression is 8° and Angie is 38 meters above the ground, what isthe straight-line distance from Angie to her car? Round to the nearest whole meter.
8. INDIRECT MEASUREMENT Kyle is at the end of a pier 30 feet above the ocean. His eye level is 3 feet above the pier. He is using binoculars to watch a whale surface. If the angle of depression of the whale is 20°, how far is the whale from Kyle’s binoculars? Round to the nearest tenth foot.
whale water level
20�Kyle’s eyes
pier3 ft
30 ft
60�?
35 ft
Z
P
W
R
D
A
C
B
T
W
R
S
F
T
L
S
© Glencoe/McGraw-Hill 378 Glencoe Geometry
Name the angle of depression or angle of elevation in each figure.
1. 2.
3. WATER TOWERS A student can see a water tower from the closest point of the soccerfield at San Lobos High School. The edge of the soccer field is about 110 feet from thewater tower and the water tower stands at a height of 32.5 feet. What is the angle ofelevation if the eye level of the student viewing the tower from the edge of the soccerfield is 6 feet above the ground? Round to the nearest tenth degree.
4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladderreaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of theladder to the roof is 55°, how far is the ladder from the base of the wall? Round youranswer to the nearest foot.
5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet on any property. Lindsay wants to determinewhether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from theflagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth foot?
6. GEOGRAPHY Stephan is standing on a mesa at the Painted Desert. The elevation ofthe mesa is about 1380 meters and Stephan’s eye level is 1.8 meters above ground. IfStephan can see a band of multicolored shale at the bottom and the angle of depressionis 29°, about how far is the band of shale from his eyes? Round to the nearest meter.
7. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34°and 48°, how far apart are the dogs to the nearest foot?
48� 34�
40 ft
6 ft
Mr. Dominguez
bluff
25�5.5 ft
36 ft
x
R
M
P
L
T
Y
R
Z
Practice Angles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Reading to Learn MathematicsAngles of Elevation and Depression
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 379 Glencoe Geometry
Less
on
7-5
Pre-Activity How do airline pilots use angles of elevation and depression?
Read the introduction to Lesson 7-5 at the top of page 371 in your textbook.
What does the angle measure tell the pilot?
Reading the Lesson
1. Refer to the figure. The two observers are looking at one another. Select the correct choice for each question.
a. What is the line of sight?(i) line RS (ii) line ST (iii) line RT (iv) line TU
b. What is the angle of elevation?(i) �RST (ii) �SRT (iii) �RTS (iv) �UTR
c. What is the angle of depression?(i) �RST (ii) �SRT (iii) �RTS (iv) �UTR
d. How are the angle of elevation and the angle of depression related?(i) They are complementary.(ii) They are congruent.(iii) They are supplementary.(iv) The angle of elevation is larger than the angle of depression.
e. Which postulate or theorem that you learned in Chapter 3 supports your answer forpart c?(i) Corresponding Angles Postulate(ii) Alternate Exterior Angles Theorem(iii) Consecutive Interior Angles Theorem(iv) Alternate Interior Angles Theorem
2. A student says that the angle of elevation from his eye to the top of a flagpole is 135°.What is wrong with the student’s statement?
Helping You Remember
3. A good way to remember something is to explain it to someone else. Suppose a classmatefinds it difficult to distinguish between angles of elevation and angles of depression. Whatare some hints you can give her to help her get it right every time?
S
Tobserver at
top of building
observeron ground R
U
© Glencoe/McGraw-Hill 380 Glencoe Geometry
Reading MathematicsThe three most common trigonometric ratios are sine, cosine, and tangent. Three other ratios are thecosecant, secant, and cotangent. The chart below shows abbreviations and definitions for all six ratios.Refer to the triangle at the right.
Use the abbreviations to rewrite each statement as an equation.
1. The secant of angle A is equal to 1 divided by the cosine of angle A.
2. The cosecant of angle A is equal to 1 divided by the sine of angle A.
3. The cotangent of angle A is equal to 1 divided by the tangent of angle A.
4. The cosecant of angle A multiplied by the sine of angle A is equal to 1.
5. The secant of angle A multiplied by the cosine of angle A is equal to 1.
6. The cotangent of angle A times the tangent of angle A is equal to 1.
Use the triangle at right. Write each ratio.
7. sec R 8. csc R 9. cot R
10. sec S 11. csc S 12. cot S
13. If sin x° � 0.289, find the value of csc x°.
14. If tan x° � 1.376, find the value of cot x°.
R
T S
ts
r
A
ca
b
B
C
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Abbreviation Read as: Ratio
sin A the sine of �A � �ac
�
cos A the cosine of �A � �bc
�
tan A the tangent of �A � �ab
�
csc A the cosecant of �A � �ac
�
sec A the secant of �A � �bc
�
cot A the cotangent of �A � �ba
�leg adjacent to �A���
leg opposite �A
hypotenuse���leg adjacent to �A
hypotenuse��leg opposite �A
leg opposite �A���leg adjacent to �A
leg adjacent to �A���
hypotenuse
leg opposite �A��
hypotenuse
Study Guide and InterventionThe Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 381 Glencoe Geometry
Less
on
7-6
The Law of Sines In any triangle, there is a special relationship between the angles ofthe triangle and the lengths of the sides opposite the angles.
Law of Sines �sin
aA
� � �sin
bB
� � �sin
cC
�
In �ABC, find b.
�sin
cC
� � �sin
bB
� Law of Sines
�sin
3045°� � �
sinb74°� m�C � 45, c � 30, m�B � 74
b sin 45° � 30 sin 74° Cross multiply.
b � �30
sisnin45
7°4°
� Divide each side by sin 45°.
b � 40.8 Use a calculator.
45�
3074�
b
B
AC
In �DEF, find m�D.
�sin
dD
� � �sin
eE
� Law of Sines
�si
2n8D
� � �sin
2458°�
d � 28, m�E � 58,
e � 24
24 sin D � 28 sin 58° Cross multiply.
sin D � �28 s
2in4
58°� Divide each side by 24.
D � sin�1 �28 s
2in4
58°� Use the inverse sine.
D � 81.6° Use a calculator.
58�
24
28
E
FD
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find each measure using the given measures of �ABC. Round angle measures tothe nearest degree and side measures to the nearest tenth.
1. If c � 12, m�A � 80, and m�C � 40, find a.
2. If b � 20, c � 26, and m�C � 52, find m�B.
3. If a � 18, c � 16, and m�A � 84, find m�C.
4. If a � 25, m�A � 72, and m�B � 17, find b.
5. If b � 12, m�A � 89, and m�B� 80, find a.
6. If a � 30, c � 20, and m�A � 60, find m�C.
© Glencoe/McGraw-Hill 382 Glencoe Geometry
Use the Law of Sines to Solve Problems You can use the Law of Sines to solvesome problems that involve triangles.
Law of SinesLet �ABC be any triangle with a, b, and c representing the measures of the sides opposite
the angles with measures A, B, and C, respectively. Then �sina
A� � �sinb
B� � �sinc
C�.
Isosceles �ABC has a base of 24 centimeters and a vertex angle of 68°. Find the perimeter of the triangle.The vertex angle is 68°, so the sum of the measures of the base angles is 112 and m�A � m�C � 56.
�sin
bB
� � �sin
aA
� Law of Sines
�sin
2468°� � �
sina56°� m�B � 68, b � 24, m�A � 56
a sin 68° � 24 sin 56° Cross multiply.
a � �24
sisnin68
5°6°
� Divide each side by sin 68°.
� 21.5 Use a calculator.
The triangle is isosceles, so c � 21.5.The perimeter is 24 � 21.5 � 21.5 or about 67 centimeters.
Draw a triangle to go with each exercise and mark it with the given information.Then solve the problem. Round angle measures to the nearest degree and sidemeasures to the nearest tenth.
1. One side of a triangular garden is 42.0 feet. The angles on each end of this side measure66° and 82°. Find the length of fence needed to enclose the garden.
2. Two radar stations A and B are 32 miles apart. They locate an airplane X at the sametime. The three points form �XAB, which measures 46°, and �XBA, which measures52°. How far is the airplane from each station?
3. A civil engineer wants to determine the distances from points A and B to an inaccessiblepoint C in a river. �BAC measures 67° and �ABC measures 52°. If points A and B are82.0 feet apart, find the distance from C to each point.
4. A ranger tower at point A is 42 kilometers north of a ranger tower at point B. A fire atpoint C is observed from both towers. If �BAC measures 43° and �ABC measures 68°,which ranger tower is closer to the fire? How much closer?
68�
b
c a
24
B
CA
Study Guide and Intervention (continued)
The Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
ExampleExample
ExercisesExercises
Skills PracticeThe Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 383 Glencoe Geometry
Less
on
7-6
Find each measure using the given measures from �ABC. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.
1. If m�A � 35, m�B � 48, and b � 28, find a.
2. If m�B � 17, m�C � 46, and c � 18, find b.
3. If m�C � 86, m�A � 51, and a � 38, find c.
4. If a � 17, b � 8, and m�A � 73, find m�B.
5. If c � 38, b � 34, and m�B � 36, find m�C.
6. If a � 12, c � 20, and m�C � 83, find m�A.
7. If m�A � 22, a � 18, and m�B� 104, find b.
Solve each �PQR described below. Round measures to the nearest tenth.
8. p � 27, q � 40, m�P � 33
9. q � 12, r � 11, m�R � 16
10. p � 29, q � 34, m�Q � 111
11. If m�P � 89, p � 16, r � 12
12. If m�Q � 103, m�P � 63, p � 13
13. If m�P � 96, m�R � 82, r � 35
14. If m�R � 49, m�Q � 76, r � 26
15. If m�Q � 31, m�P � 52, p � 20
16. If q � 8, m�Q � 28, m�R � 72
17. If r � 15, p � 21, m�P � 128
© Glencoe/McGraw-Hill 384 Glencoe Geometry
Find each measure using the given measures from �EFG. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.
1. If m�G � 14, m�E � 67, and e � 14, find g.
2. If e � 12.7, m�E � 42, and m�F � 61, find f.
3. If g � 14, f � 5.8, and m�G � 83, find m�F.
4. If e � 19.1, m�G � 34, and m�E � 56, find g.
5. If f � 9.6, g � 27.4, and m�G � 43, find m�F.
Solve each �STU described below. Round measures to the nearest tenth.
6. m�T � 85, s � 4.3, t � 8.2
7. s � 40, u � 12, m�S � 37
8. m�U � 37, t � 2.3, m�T � 17
9. m�S � 62, m�U � 59, s � 17.8
10. t � 28.4, u � 21.7, m�T � 66
11. m�S � 89, s � 15.3, t � 14
12. m�T � 98, m�U � 74, u � 9.6
13. t � 11.8, m�S � 84, m�T � 47
14. INDIRECT MEASUREMENT To find the distance from the edge of the lake to the tree on the island in the lake, Hannah set up atriangular configuration as shown in the diagram. The distance from location A to location B is 85 meters. The measures of the angles at A and B are 51° and 83°, respectively. What is the distancefrom the edge of the lake at B to the tree on the island at C?
A
C
B
Practice The Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
Reading to Learn MathematicsThe Law of Sines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
© Glencoe/McGraw-Hill 385 Glencoe Geometry
Less
on
7-6
Pre-Activity How are triangles used in radio astronomy?
Read the introduction to Lesson 7-6 at the top of page 377 in your textbook.
Why might several antennas be better than one single antenna whenstudying distant objects?
Reading the Lesson
1. Refer to the figure. According to the Law of Sines, which of the following are correct statements?
A. �sinm
M� � �sinn
N� � �sinp
P� B. �sin
Mm
� � �si
Nn n� � �
sinP
p�
C. �co
ms M� � �
cosn
N� � �
cops P� D. �
sinm
M� � �
sinn
N� � �
sinp
P�
E. (sin M)2 � (sin N)2 � (sin P)2 F. �sin
pP
� � �sin
mM
� � �sin
nN
�
2. State whether each of the following statements is true or false. If the statement is false,explain why.
a. The Law of Sines applies to all triangles.
b. The Pythagorean Theorem applies to all triangles.
c. If you are given the length of one side of a triangle and the measures of any twoangles, you can use the Law of Sines to find the lengths of the other two sides.
d. If you know the measures of two angles of a triangle, you should use the Law of Sinesto find the measure of the third angle.
e. A friend tells you that in triangle RST, m�R � 132, r � 24 centimeters, and s � 31centimeters. Can you use the Law of Sines to solve the triangle? Explain.
Helping You Remember
3. Many students remember mathematical equations and formulas better if they can statethem in words. State the Law of Sines in your own words without using variables ormathematical symbols.
P
M Np
mn
© Glencoe/McGraw-Hill 386 Glencoe Geometry
IdentitiesAn identity is an equation that is true for all values of the variable for which both sides are defined. One way to verify an identity is to use a right triangle and the definitions fortrigonometric functions.
Verify that (sin A)2 � (cos A)2 � 1 is an identity.
(sin A)2 � (cos A)2 � ��ac��2 � ��
bc��2
� �a2 �
cb2
� � �cc
2
2� � 1
To check whether an equation may be an identity, you can testseveral values. However, since you cannot test all values, youcannot be certain that the equation is an identity.
Test sin 2x � 2 sin x cos x to see if it could be an identity.
Try x � 20. Use a calculator to evaluate each expression.
sin 2x � sin 40 2 sin x cos x � 2 (sin 20)(cos 20)� 0.643 � 2(0.342)(0.940)
� 0.643
Since the left and right sides seem equal, the equation may be an identity.
Use triangle ABC shown above. Verify that each equation is an identity.
1. �csoins
AA
� � �tan1
A� 2. �tsainn
BB
� � �co1s B�
3. tan B cos B � sin B 4. 1 � (cos B)2 � (sin B)2
Try several values for x to test whether each equation could be an identity.
5. cos 2x � (cos x)2 � (sin x)2 6. cos (90 � x) � sin x
B
A C
ca
b
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-67-6
Example 1Example 1
Example 2Example 2
Study Guide and InterventionThe Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 387 Glencoe Geometry
Less
on
7-7
The Law of Cosines Another relationship between the sides and angles of any triangleis called the Law of Cosines. You can use the Law of Cosines if you know three sides of atriangle or if you know two sides and the included angle of a triangle.
Let �ABC be any triangle with a, b, and c representing the measures of the sides opposite Law of Cosines the angles with measures A, B, and C, respectively. Then the following equations are true.
a2 � b2 � c2 � 2bc cos A b2 � a2 � c2 � 2ac cos B c2 � a2 � b2 � 2ab cos C
In �ABC, find c.c2 � a2 � b2 � 2ab cos C Law of Cosines
c2 � 122 � 102 � 2(12)(10)cos 48° a � 12, b � 10, m�C � 48
c � �122 �� 102 �� 2(12)�(10)co�s 48°� Take the square root of each side.
c � 9.1 Use a calculator.
In �ABC, find m�A.a2 � b2 � c2 � 2bc cos A Law of Cosines
72 � 52 � 82 � 2(5)(8) cos A a � 7, b � 5, c � 8
49 � 25 � 64 � 80 cos A Multiply.
�40 � �80 cos A Subtract 89 from each side.
�12� � cos A Divide each side by �80.
cos�1 �12� � A Use the inverse cosine.
60° � A Use a calculator.
Find each measure using the given measures from �ABC. Round angle measuresto the nearest degree and side measures to the nearest tenth.
1. If b � 14, c � 12, and m�A � 62, find a.
2. If a � 11, b � 10, and c � 12, find m�B.
3. If a � 24, b � 18, and c � 16, find m�C.
4. If a � 20, c � 25, and m�B � 82, find b.
5. If b � 18, c � 28, and m�A � 59, find a.
6. If a � 15, b � 19, and c � 15, find m�C.
58
7 CB
A
48�12 10
c
C
BA
Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 388 Glencoe Geometry
Use the Law of Cosines to Solve Problems You can use the Law of Cosines tosolve some problems involving triangles.
Let �ABC be any triangle with a, b, and c representing the measures of the sides opposite the Law of Cosines angles with measures A, B, and C, respectively. Then the following equations are true.
a2 � b2 � c2 � 2bc cos A b2 � a2 � c2 � 2ac cos B c2 � a2 � b2 � 2ab cos C
Ms. Jones wants to purchase a piece of land with the shape shown. Find the perimeter of the property.Use the Law of Cosines to find the value of a.
a2 � b2 � c2 � 2bc cos A Law of Cosines
a2 � 3002 � 2002 � 2(300)(200) cos 88° b � 300, c � 200, m�A � 88
a � �130,0�00 ��120,0�00 cos� 88°� Take the square root of each side.
� 354.7 Use a calculator.
Use the Law of Cosines again to find the value of c.
c2 � a2 � b2 � 2ab cos C Law of Cosines
c2 � 354.72 � 3002 � 2(354.7)(300) cos 80° a � 354.7, b � 300, m�C � 80
c � �215,8�12.09� � 21�2,820� cos 8�0°� Take the square root of each side.
� 422.9 Use a calculator.
The perimeter of the land is 300 � 200 � 422.9 � 200 or about 1223 feet.
Draw a figure or diagram to go with each exercise and mark it with the giveninformation. Then solve the problem. Round angle measures to the nearest degreeand side measures to the nearest tenth.
1. A triangular garden has dimensions 54 feet, 48 feet, and 62 feet. Find the angles at eachcorner of the garden.
2. A parallelogram has a 68° angle and sides 8 and 12. Find the lengths of the diagonals.
3. An airplane is sighted from two locations, and its position forms an acute triangle withthem. The distance to the airplane is 20 miles from one location with an angle ofelevation 48.0°, and 40 miles from the other location with an angle of elevation of 21.8°.How far apart are the two locations?
4. A ranger tower at point A is directly north of a ranger tower at point B. A fire at point Cis observed from both towers. The distance from the fire to tower A is 60 miles, and thedistance from the fire to tower B is 50 miles. If m�ACB � 62, find the distance betweenthe towers.
200 ft
300 ft
300 ft
88�
80�ca
Study Guide and Intervention (continued)
The Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
ExampleExample
ExercisesExercises
Skills PracticeThe Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 389 Glencoe Geometry
Less
on
7-7
In �RST, given the following measures, find the measure of the missing side.
1. r � 5, s � 8, m�T � 39
2. r � 6, t � 11, m�S � 87
3. r � 9, t � 15, m�S � 103
4. s � 12, t � 10, m�R � 58
In �HIJ, given the lengths of the sides, find the measure of the stated angle to thenearest tenth.
5. h � 12, i � 18, j � 7; m�H
6. h � 15, i � 16, j � 22; m�I
7. h � 23, i � 27, j � 29; m�J
8. h � 37, i � 21, j � 30; m�H
Determine whether the Law of Sines or the Law of Cosines should be used first tosolve each triangle. Then solve each triangle. Round angle measures to the nearestdegree and side measures to the nearest tenth.
9. 10.
11. a � 10, b � 14, c �19 12. a � 12, b � 10, m�C � 27
Solve each �RST described below. Round measures to the nearest tenth.
13. r � 12, s � 32, t � 34
14. r � 30, s � 25, m�T � 42
15. r � 15, s � 11, m�R � 67
16. r � 21, s � 28, t � 30
M
L N
�86�
52
24
B
A C
c
66�
33
19
© Glencoe/McGraw-Hill 390 Glencoe Geometry
In �JKL, given the following measures, find the measure of the missing side.
1. j � 1.3, k � 10, m�L � 77
2. j � 9.6, � � 1.7, m�K � 43
3. j � 11, k � 7, m�L � 63
4. k � 4.7, � � 5.2, m�J � 112
In �MNQ, given the lengths of the sides, find the measure of the stated angle tothe nearest tenth.
5. m � 17, n � 23, q � 25; m�Q
6. m � 24, n � 28, q � 34; m�M
7. m � 12.9, n � 18, q � 20.5; m�N
8. m � 23, n � 30.1, q � 42; m�Q
Determine whether the Law of Sines or the Law of Cosines should be used first tosolve �ABC. Then sole each triangle. Round angle measures to the nearest degreeand side measure to the nearest tenth.
9. a � 13, b � 18, c � 19 10. a � 6, b � 19, m�C � 38
11. a � 17, b � 22, m�B � 49 12. a � 15.5, b � 18, m�C � 72
Solve each �FGH described below. Round measures to the nearest tenth.
13. m�F � 54, f � 12.5, g � 11
14. f �20, g � 23, m�H � 47
15. f � 15.8, g � 11, h � 14
16. f � 36, h � 30, m�G � 54
17. REAL ESTATE The Esposito family purchased a triangular plot of land on which theyplan to build a barn and corral. The lengths of the sides of the plot are 320 feet, 286 feet,and 305 feet. What are the measures of the angles formed on each side of the property?
Practice The Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
Reading to Learn MathematicsThe Law of Cosines
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
© Glencoe/McGraw-Hill 391 Glencoe Geometry
Less
on
7-7
Pre-Activity How are triangles used in building design?
Read the introduction to Lesson 7-7 at the top of page 385 in your textbook.
What could be a disadvantage of a triangular room?
Reading the Lesson1. Refer to the figure. According to the Law of Cosines, which
statements are correct for �DEF?
A. d2 � e2 � f 2 � ef cos D B. e2 � d2 � f 2 � 2df cos E
C. d2 � e2 � f 2 � 2ef cos D D. f 2 � d2 � e2 � 2ef cos F
E. f2 � d2 � e2 � 2de cos F F. d2 � e2 � f 2
G. �sin
dD
� � �sin
eE
� � �sin
fF
� H. d � �e2 � f�2 � 2e�f cos �D�
2. Each of the following describes three given parts of a triangle. In each case, indicatewhether you would use the Law of Sines or the Law of Cosines first in solving a trianglewith those given parts. (In some cases, only one of the two laws would be used in solvingthe triangle.)
a. SSS b. ASA
c. AAS d. SAS
e. SSA
3. Indicate whether each statement is true or false. If the statement is false, explain why.
a. The Law of Cosines applies to right triangles.
b. The Pythagorean Theorem applies to acute triangles.
c. The Law of Cosines is used to find the third side of a triangle when you are given themeasures of two sides and the nonincluded angle.
d. The Law of Cosines can be used to solve a triangle in which the measures of the threesides are 5 centimeters, 8 centimeters, and 15 centimeters.
Helping You Remember4. A good way to remember a new mathematical formula is to relate it to one you already
know. The Law of Cosines looks somewhat like the Pythagorean Theorem. Both formulasmust be true for a right triangle. How can that be?
D
dE
e
F
f
© Glencoe/McGraw-Hill 392 Glencoe Geometry
Spherical TrianglesSpherical trigonometry is an extension of plane trigonometry.Figures are drawn on the surface of a sphere. Arcs of great circles correspond to line segments in the plane. The arcs of three great circles intersecting on a sphere form a spherical triangle. Angles have the same measure as the tangent lines drawn to each great circle at the vertex. Since the sides are arcs, they too can be measured in degrees.
Solve the spherical triangle given a � 72�,b � 105�, and c � 61�.
Use the Law of Cosines.
0.3090 � (–0.2588)(0.4848) � (0.9659)(0.8746) cos Acos A � 0.5143
A � 59°
�0.2588 � (0.3090)(0.4848) � (0.9511)(0.8746) cos Bcos B � �0.4912
B � 119°
0.4848 � (0.3090)(–0.2588) � (0.9511)(0.9659) cos Ccos C � 0.6148
C � 52°
Check by using the Law of Sines.
�ssiinn
75
29
°°� � �
ssiinn
11
01
59
°°� � �
ssiinn
65
12
°°� � 1.1
Solve each spherical triangle.
1. a � 56°, b � 53°, c � 94° 2. a � 110°, b � 33°, c � 97°
3. a � 76°, b � 110°, C � 49° 4. b � 94°, c � 55°, A � 48°
A
C
B
c
ba
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-77-7
The sum of the sides of a spherical triangle is less than 360°.The sum of the angles is greater than 180° and less than 540°.The Law of Sines for spherical triangles is as follows.
�ssiinn
Aa
� � �ssiinn
Bb
� � �ssiinn
Cc
�
There is also a Law of Cosines for spherical triangles.
cos a � cos b cos c � sin b sin c cos A
cos b � cos a cos c � sin a sin c cos B
cos c � cos a cos b � sin a sin b cos C
ExampleExample
Chapter 7 Test, Form 177
© Glencoe/McGraw-Hill 393 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of each question.
1. Find the geometric mean between 20 and 5.A. 100 B. 50 C. 12.5 D. 10
2. Find x in �ABC.A. 8 B. 10C. �20� D. 64
3. Find x in �PQR.A. 13 B. 15C. 16 D. �60�
4. Find x in �STU.A. 2 B. 8C. �32� D. �514�
5. Which set of measures could represent the sides of a right triangle?A. 2, 3, 4 B. 7, 11, 14C. 8, 10, 12 D. 9, 12, 15
6. Find x in �DEF.A. 6 B. 6�2�C. 6�3� D. 12
7. Find y in �XYZ.A. 7.5�3� B. 15�3�C. 15 D. 30
8. The length of the sides of a square is 10 meters. Find the length of thediagonal of the square.A. 10 m B. 10�2� mC. 10�3� m D. 20 m
9. Find x in �HJK.A. 5�2� B. 5�3�C. 10 D. 15
10. Find x in �ABC.A. 25 B. 25�2�C. 25�3� D. 100
A C
B
60� 30�50
x
H J
K60�
30�
5x
X Y
Z
15��245�
45�
y
D E
F
6
6x
S T
U
15
17x
P Q
R
12
5 x
A B
C 4
16x
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 394 Glencoe Geometry
Chapter 7 Test, Form 1 (continued)77
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
11. In �QRS, �R is a right angle. Which is the ratio for the tangent of �S?
A. B.
C. D.
12. Find cos A in �ABC.
A. �274� B. �2
75�
C. �22
54� D. �
22
45�
13. Find x to the nearest tenth.A. 7.3 B. 17.3C. 18.4 D. 47.1
14. Find the angle of elevation of the sun when a pole 25 feet tall casts a shadow42 feet long.A. 30.8° B. 36.5° C. 53.5° D. 59.2°
15. Which is the angle of depression in the figure at the right?A. �AOT B. �AOBC. �TOB D. �BTO
16. Find y in �XYZ to the nearest tenth if m�Y � 36, m�X � 49, and x � 12.A. 0.04 B. 9.35 C. 14.80 D. 15.41
17. To find the distance between two points A and B on opposite sides of a river, a surveyor measures the distance from A to C as 200 feet, m�A � 72, and m�B � 37. Find the distance from A to B. Round your answer to the nearest tenth.A. 77.4 ft B. 201.2 ft C. 250.4 ft D. 314.2 ft
18. In �ABC, a � 12, b � 8, and m�A � 40. Find m�B to the nearest tenth.A. 25.4 B. 56.3 C. 59.3 D. 74.6
19. Find the third side of a triangular garden if two sides are 8 feet and 12 feetand the included angle has a measure 50.A. 7.8 ft B. 9.2 ft C. 14.4 ft D. 146.3 ft
20. In �DEF, d � 20, e � 25, and f � 30. Find m�F to the nearest tenth.A. 82.8 B. 75.5 C. 55.8 D. 47.2
Bonus In �ABC, a � 50, b � 48, and c � 40. Find m�A to the nearest tenth.
A B
C
AO
B T
67� 20
x
C A
B21
72
75
measure of leg opposite �S����measure of leg adjacent to �S
measure of leg opposite �S����measure of hypotenuse
measure of hypotenuse����measure of leg opposite �S
measure of leg adjacent to �S����measure of hypotenuse
B:
NAME DATE PERIOD
Chapter 7 Test, Form 2A77
© Glencoe/McGraw-Hill 395 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of each question.
1. Find the geometric mean between 7 and 12.A. 5 B. 9.5C. �19� D. �84�
2. In �PQR, RS � 4 and QS � 6. Find PS.A. 2 B. 5C. �10� D. �24�
3. Find x.A. �18� B. �14�C. 4.5 D. 3
4. Find y.A. 12 B. 11C. 9 D. 2
5. Find the length of the hypotenuse of a right triangle whose legs measure 5 and 7.A. 12 B. �24�C. �35� D. �74�
6. Find x.A. 3 B. 4C. 4�3� D. 2�5�
7. Which of the following could represent sides of a right triangle?A. 9, 40, 41 B. 8, 30, 31C. 7, 8, 15 D. �2�, �3�, �6�
8. Find c.A. 7 B. 7�2�C. 7�3� D. 14
9. Find the perimeter of a square to the nearest tenth if the length of itsdiagonal is 12 inches.A. 8.5 in. B. 33.9 in.C. 48 in. D. 67.9 in.
10. Find x.A. 4 B. 4�2�C. 4�3� D. 8�3�
x 88
60�
45�
7c
x66
8
3
y 6
2 7
x
P Q
SR 4
6
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 396 Glencoe Geometry
Chapter 7 Test, Form 2A (continued)77
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
11. Find x to the nearest tenth.A. 5.8 B. 5.9C. 8.1 D. 17.3
12. In right triangle ABC, a � 12, b � 9, and c � 15. Find tan �B.
A. �43� B. �
54� C. �
34� D. �
35�
13. Find x to the nearest tenth of a degree.A. 56.3 B. 45C. 33.7 D. 29.1
14. If a 20-foot ladder makes a 65° angle with the ground, how many feet up awall will it reach? Round your answer to the nearest tenth.A. 8.5 ft B. 10 ft C. 18.1 ft D. 42.9 ft
15. A ship’s sonar finds that the angle of depression to a wreck on the bottom ofthe ocean is 12.5°. If a point on the ocean floor is 60 meters directly below theship, how many meters, to the nearest tenth, is it from that point on theocean floor to the wreck?A. 277.2 m B. 270.6 m C. 61.5 m D. 13.3 m
16. To the nearest tenth of a degree, find the angle of elevation of the sun if abuilding 100 feet tall casts a shadow 150 feet long.A. 60° B. 48.2° C. 41.8° D. 33.7°
17. When the sun’s angle of elevation is 73°, a tree tilted at an angle of 5° from the vertical casts a 20-foot shadow on the ground. Find the length of the tree to the nearest tenth.A. 6.3 ft B. 19.2 ftC. 51.1 ft D. 219.4 ft
18. In �CDE, m�C � 52, m�D � 17, and e � 28.6. Find c to the nearest tenth.A. 77.1 B. 49.1 C. 24.1 D. 18.4
19. In �PQR, p � 56, r � 17, and m�Q � 110. Find q to the nearest tenth.A. 4076.2 B. 63.8 C. 52.6 D. 3.1
20. Pete is building a kite using the dimensions given in the figure at the right. Find the measure of the angle the 2-foot edge makes with the 3-foot edge.A. 104.5 B. 85.2C. 60 D. 14.5
Bonus From a window 20 feet above the ground, the angle of elevation to the top of another building is 35°. The distance between the buildings is 52 feet. Find the height of the building to the nearest tenth of a foot.
2 ft 2 ft
3 ft3 ft4 ft
73�
5�
20-foot shadow
✹
95
x
10
36�
x
B:
NAME DATE PERIOD
Chapter 7 Test, Form 2B77
© Glencoe/McGraw-Hill 397 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of each question.
1. Find the geometric mean between 9 and 11.A. �99� B. �20�C. 10 D. 2
2. In �PQR, RS � 5 and QS � 8. Find PS.A. 3 B. 6.5C. �13� D. �40�
3. Find x.A. 5.5 B. �11�C. �24� D. �33�
4. Find y.A. 4 B. 5C. 8 D. 9
5. Find the length of the hypotenuse of a right triangle whose legs measure 6 and 5.A. 11 B. �11�C. �30� D. �61�
6. Find x.A. �39� B. 6C. 5�3� D. 5
7. Which of the following could represent sides of a right triangle?
A. �34�, 1, �
54� B. �3�, �5�, �15�
C. 7, 17, 24 D. 8, 15, 16
8. Find c.A. 18 B. 9�3�C. 9�2� D. 9
9. Find the perimeter of a square to the nearest tenth if the length of itsdiagonal is 16 millimeters.A. 11.3 mm B. 45.3 mmC. 90.5 mm D. 128.0 mm
10. Find x.A. 6 B. 6�2�C. 6�3� D. 12�3�
x 1212
60�
45�
9c
x8
8
10
y
4
6
x
83
P Q
SR 5
8
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 398 Glencoe Geometry
Chapter 7 Test, Form 2B (continued)77
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
11. Find x.A. 8.0 B. 8.9C. 10.4 D. 10.8
12. In right triangle ABC, a � 14, b � 48, and c � 50. Find tan �A.
A. �274� B. �2
75� C. �
22
45� D. �
274�
13. Find x to the nearest tenth of a degree.A. 56.9 B. 54.5C. 33.1 D. 28.6
14. If a 24-foot ladder makes a 58° angle with the ground, how many feet up awall will it reach? Round your answer to the nearest tenth.A. 38.4 ft B. 20.8 ft C. 20.4 ft D. 12.7 ft
15. A ship’s sonar finds that the angle of depression to a wreck on the bottom ofthe ocean is 13.2°. If a point on the ocean floor is 75 meters directly below theship, how many meters, to the nearest tenth, is it from that point on theocean floor to the wreck?A. 328.4 m B. 319.8 m C. 77.0 m D. 17.6 m
16. To the nearest tenth of a degree, find the angle of elevation of the sun if abuilding 125 feet tall casts a shadow 196 feet long.A. 63.8° B. 50.4° C. 39.6° D. 32.5°
17. When the sun’s angle of elevation is 76°, a tree tilted at an angle of 4° from the vertical casts a 18-foot shadow on the ground. Find the length of the tree, to the nearest tenth.A. 250.4 ft B. 56.5 ftC. 17.7 ft D. 4.6 ft
18. In �ABC, m�A � 46, m�B � 105, and c � 19.8. Find a to the nearest tenth.A. 29.4 B. 28.5 C. 15.7 D. 14.7
19. In �LMN, l � 42, m � 61, and m�N � 108. Find n to the nearest tenth.A. 7068.4 B. 84.1 C. 79.2 D. 24.7
20. Josephine is planning a triangular garden. If the lengths of the sides are 50 feet, 80 feet, and 100 feet, what is the measure of the largest angle?A. 7.9° B. 82.1° C. 89.9° D. 97.9°
Bonus From a window 24 feet above the ground, the angle of elevation to the top of another building is 38°. The distance between the buildings is 63 feet. Find the height of the building to the nearest tenth of a foot.
76�
4�
18-foot shadow
✹
x
116
x
12
42�
B:
NAME DATE PERIOD
Chapter 7 Test, Form 2C77
© Glencoe/McGraw-Hill 399 Glencoe Geometry
Ass
essm
ents
1. Find the geometric mean between 2�5� and 5�2�.
For Questions 2–5, find x.
2. 3.
4. 5.
6. Determine whether �ABC is a right triangle. Explain your answer.
7. Find x.
8. In parallelogram ABCD, AD � 4 and m�D � 60. Find AF.
9. Find x and y.
10. Find x to the nearest tenth.
11. An A-frame house is 40 feet high and 30 feet wide. Find the measure of theangle, to the nearest tenth of a degree,that the roof makes with the floor.
12. A 30-foot tree casts a 12-foot shadow. Find the angle ofelevation of the sun to the nearest tenth of a degree.
30 ftx
40 ft
9.218�
x
4��3
60�
30�
x
y
A D
CF
B
22
x
x
y
O
C
A
B
x 2020
20
60
20x
2 12
x
6
4x
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 400 Glencoe Geometry
Chapter 7 Test, Form 2C (continued)77
13. A boat is 1000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is 15°, how tall is the cliff? Round your answer to the nearest tenth.
14. A plane flying at an altitude of 10,000 feet begins descendingwhen the end of the runway is below a point 50,000 feet away.Find the angle of descent (depression) to the nearest tenth of adegree.
15. Find x to the nearest tenth.
16. Find x to the nearest tenth of a degree.
17. A tree grew at a 3° slant from the vertical. At a point 50 feet from the tree, the angle of elevation to the top of the tree is 17°. Find the length of the tree to the nearest tenth of a foot.
18. Find x to the nearest tenth of a degree.
19. In �XYZ, m�X � 152, y � 15, and z � 19. Find x to thenearest tenth.
20. To approximate the length of a pond, a surveyor walks 400 meters from point L to point K, then turns and walks 220 meters from point Kto point E. If m �LKE � 110, find the length LE of the pond to the nearest tenth of a meter.
Bonus Find x.��6
5x
L
400 m 220 m110�
E
K
11
75x
17� 93�
50 ft
x
157
23� x
52�
26
37�x
1000 m
NAME DATE PERIOD
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 7 Test, Form 2D77
© Glencoe/McGraw-Hill 401 Glencoe Geometry
Ass
essm
ents
1. Find the geometric mean between 3�6� and 5�6�.
For Questions 2–5, find x.
2. 3.
4. 5.
6. Determine whether �ABC is a right triangle. Explain your answer.
7. Find x.
8. In parallelogram ABCD, AD � 14 and m�D � 60. Find AF.
9. Find x and y.
10. Find x to the nearest tenth.
11. An A-frame house is 45 feet high and 32 feet wide. Find the measure of theangle, to the nearest tenth of a degree,that the roof makes with the floor.
12. A 38-foot tree casts a 16-foot shadow. Find the angle ofelevation of the sun to the nearest tenth of a degree.
32 ftx
45 ft
16�8.3
x
8��330�
60�
x
y
A D
CF
B
30
x
x
y
O
C
A
B
x 2424
24
80
30x
103
x
12
8x
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 402 Glencoe Geometry
Chapter 7 Test, Form 2D (continued)77
13. A boat is 2000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is 10°, how tall is the cliff? Round your answer to the nearest tenth.
14. A plane flying at an altitude of 10,000 feet begins descendingwhen the end of the runway is below a point 60,000 feet away.Find the angle of descent (depression) to the nearest tenth of adegree.
15. Find x to the nearest tenth.
16. Find x to the nearest tenth of a degree.
17. A tree grew at a 3° slant from the vertical. At a point 60 feet from the tree, the angle of elevation to the top of the tree is 14°. Find the length of the tree to the nearest tenth of a foot.
18. Find x to the nearest tenth of a degree.
19. In �XYZ, m�X � 156, y � 18, and z � 21. Find x to thenearest tenth.
20. To approximate the length of a pond,a surveyor walks 420 meters from point L to point K, then turns and walks 280 meters from point K to point E. If m�LKE � 125, find the length LE of the pond to the nearest tenth of a meter.
Bonus Find x.
2��15
16x
L
420 m 280 m125�
E
K
8 9
16x
14� 93�
60 ft
x
38�
17
6
x
68�
42�
23 x
2000 m
NAME DATE PERIOD
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 7 Test, Form 377
© Glencoe/McGraw-Hill 403 Glencoe Geometry
Ass
essm
ents
1. Find the geometric mean between �29� and �
39�.
2. Find x in �PQR.
3. Find x in �XYZ.
4. If the length of one leg of a right triangle is three times thelength of the other and the hypotenuse is 20, find the length ofthe shorter leg.
5. Find the length of the altitude to the hypotenuse of a righttriangle with legs of length 3 and 4.
6. Find x.
7. Richmond is 200 kilometers due east of Teratown and Hamiltonis 150 kilometers directly north of Teratown. Find the shortestdistance in kilometers between Hamilton and Richmond.
8. Is 48, 55, 73 a Pythagorean triple? Show why or why not.
9. Find the perimeter of this square.
10. Find the perimeter of rectangle ABCD.
11. Find x and y.
12. �ABC is a 30°-60°-90° triangle with right angle A and withA�C� as the longer leg. Find the coordinates of C if A(�4, �2)and B(�4, 6).
1530�
60�x
y
12
60�
A B
CD
36��
17
89 3.5
x
x � 4 xXW Z
Y
21��
65
2x
Q
P
R
S
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 404 Glencoe Geometry
Chapter 7 Test, Form 3 (continued)77
13. If A�B� || C�D�, find x and the length of C�D�.
14. The angle of elevation from a point on the street to the top of abuilding is 29°. The angle of elevation from another point onthe street, 50 feet farther away from the building, to the top ofthe building is 25°. To the nearest foot, how tall is the building?
15. The angle of depression from the top of a flagpole on top of alighthouse to a boat on the ocean is 37°, while the angle ofdepression from the bottom of the flagpole to the boat is 36.8°.If the boat is 1 mile away from shore and the lighthouse isright on the edge of the shore, how tall is the flagpole? Roundyour answer to the nearest foot.
16. In �JKL, m�J � 26.8, m�K � 19, and k � 17. Find � to thenearest tenth.
17. Solve �PQR for r � 22, p � 51, and m�Q � 96. Round answersto the nearest tenth.
18. Don hit a golf ball 250 yards toward the hole. However, due tothe wind, his drive is 5° off course. If the angle between the holeand where the ball lands is 97°, how far is it from where Donhit the ball to the hole? Round to the nearest tenth of a yard.
19. In �HJK, m�H � 32, k � 8, and h � 7. Find m�K. Roundyour answer(s) to the nearest tenth of a degree.
20. The distance from Albany to Bethany is 75 miles and fromBethany to Celina 105 miles. If the roads from Bethany toAlbany and from Bethany to Celina make an 87° angle, what isthe distance from Albany to Celina? Round to the nearest tenth.
Bonus A 50-foot vertical pole that stands on a hillside makes anangle of 10° with the horizontal. Two guy wires extendfrom the top of the pole to points on the hill 60 feet uphilland downhill from its base. Find the length of each guywire to the nearest tenth.
12
10
60�45�x
A B
C D
NAME DATE PERIOD
13.
14.
15.
16.
17.
18.
19.
20.
B:
Chapter 7 Open-Ended Assessment77
© Glencoe/McGraw-Hill 405 Glencoe Geometry
Ass
essm
ents
Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.
1. If the geometric mean between 10 and x is 6, what is x? Show how youobtained your answer.
2.
a. Max used the following equations to find x in �PQR. Is Max correct?Why or why not?
�2x� � �8
x�
x2 � 2 � 8x2 � 16x � 4
b. For �PRQ to be a right angle, what would the measure of P�S� have to be?
c. Is �PRS a 45°-45°-90° triangle? How do you know?
3. To solve for x in a triangle, when would you use sin and when would youuse sin�1? Give an example for each type of situation.
4. Draw a diagram showing where the angles of elevation and depressionare. How are the measures of these angles related?
5. Draw an obtuse triangle and label the vertices and the measures of twoangles and the length of one side. Explain how to solve the triangle.
6. Irina is solving �ABC. She plans to first use the Law of Sines to find twoof the angles. Is Irina’s plan a good one? Why or why not?
4
15
12
A C
B
8 260�
x
P Q
R
S
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 406 Glencoe Geometry
Chapter 7 Vocabulary Test/Review77
Choose from the terms above to complete each sentence.
1. The square root of the product of two numbers is the of the numbers.
2. A group of three whole numbers that satisfy the equation a2 � b2 � c2, where c is the greatest number, is called a(n)
.
3. A ratio of the lengths of two sides of a right triangle is calleda(n) .
4. An angle between the line of sight and the horizontal when anobserver looks upward is called a(n) .
5. An angle between the line of sight and the horizontal when anobserver looks downward is called a(n) .
6. Three commonly used trigonometric ratios are the ,, and .
7. For �ABC, the says �sina
A� � �
sinb
B� � �
sinc
C�.
8. For �ABC, the says a2 � b2 � c2 � 2bc cos A.
9. The reciprocal of the sine is called the .
10. The reciprocal of the cosine is called the .
Define each term.
11. solving a triangle
12. Pythagorean Theorem
?
?
?
?
???
?
?
?
?
? 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
ambiguous caseangle of depressionangle of elevationcosecantcosine
geometric meanLaw of CosinesLaw of SinesPythagorean identityPythagorean triple
reciprocal identitiessecantsinesolving a triangle
tangenttrigonometric identitytrigonometric ratiotrigonometry
NAME DATE PERIOD
SCORE
Chapter 7 Quiz (Lessons 7–1 and 7–2)
77
© Glencoe/McGraw-Hill 407 Glencoe Geometry
Ass
essm
ents
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
1. Find the geometric mean between 12 and 16.
For Questions 2 and 3, find x and y.
2. 3.
4. Find x.
5. Do 19, 15, and 13 form a Pythagorean triple? Why or why not?
4
11
x
8
9
xy
5 12
xy
Chapter 7 Quiz (Lessons 7–3 and 7–4)
77
1.
2.
3.
4.
5.
6.
7.
8.
9.
For Questions 1 and 2, find x.
1. 2.
For Questions 3 and 4, find x to the nearest tenth.
3. 4.
5. A rectangle has a diagonal 20 inches long that forms angles of60° and 30° with the sides. Find the perimeter of the rectangle.
6. Find sin 52°. Round to the nearest ten-thousandth.
7. If cos A � 0.8945, find �A to the nearest tenth of a degree.
8. The distance along a hill is 24 feet. If the land slopes uphill atan angle of 8°, find the vertical distance from the top to thebottom of the hill.
9. A surveyor is standing on horizontal ground level with thebase of a skyscraper. The angle formed by the line segmentfrom his position to the top of the skyscraper is 31°. Theheight of the building is 1200 feet. Find the distance from thebuilding to the surveyor to the nearest foot.
1731�
x
13
11
x
60� 30�
6x
45�
6x
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 408 Glencoe Geometry
Chapter 7 Quiz (Lessons 7–5 and 7–6)
77
1.
2.
3.
4.
5.
1. Name the angle of elevation in the figure.
2. Find x to the nearest tenth.
3. Solve �ABC. Round your answers to the nearest tenth.
4. A triangular lot has 500 feet of frontage along a river. Theother two sides make angles of 48° and 75° with the riverfrontside. Find the length of the shortest side to the nearest foot.
5. STANDARDIZED TEST PRACTICE A squirrel 37 feet up in atree sees a dog 29 feet from the base of the tree. Find themeasure of the angle of depression to the nearest tenth of adegree.A. 38.4 B. 51.9 C. 45.0 D. 128.1
49�
11 18
A C
B
22�76�
10x
P
S
Q
R
NAME DATE PERIOD
SCORE
Chapter 7 Quiz (Lesson 7–7)
77
1.
2.
3.
4.
5.
For Questions 1 and 2, find x to the nearest tenth.
1. 2.
3. Solve �RST. Round your answers to the nearest degree.
4. A hiker is 6 miles from a tower and 8 miles from the lodge.She estimates that the angle between her path to the towerand her path to the lodge is 42°. Find the distance from thetower to the lodge to the nearest tenth of a mile.
5. STANDARDIZED TEST PRACTICE For �ABC, find a to thenearest tenth if m�A � 96, b � 41, and c � 50.A. 66.3 B. 67.9 C. 4395.3 D. 4609.6
48 61
76T S
R
6x
18
15
82�
x
32
23
NAME DATE PERIOD
SCORE
Chapter 7 Mid-Chapter Test (Lessons 7–1 through 7–4)
77
© Glencoe/McGraw-Hill 409 Glencoe Geometry
Ass
essm
ents
1. Find the geometric mean between 7 and 9.A. �63� B. 16 C. 8 D. 2
2. Find x.
A. �216� B.
C. 6�55� D.
3. Find sin C.
A. �2� B.
C. D.
4. Find x to the nearest tenth.A. 14 B. 18.4C. 21.1 D. 32.2
5. Find y to the nearest tenth of a degree.A. 144.9 B. 60.0C. 44.7 D. 35.1
27
19y
28
49�x
�23��5
�23���2�
�2��5
5��2
��23B C
A
�23��5
�2��5
924
24
x
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
Part II
For Questions 6–8, find x and y.
6. 7.
8.
9. Do 56, 90, 106 form a Pythagorean triple? Why or why not?
10. Guy wires 80 feet long support a 65-foot tall telephone pole. Tothe nearest tenth of a degree, what angle will the wires makewith the ground?
30�
60�
45�
8��3
yx
60� 45�
20 yx
3
6 y
x
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 410 Glencoe Geometry
Chapter 7 Cumulative Review(Chapters 1–7)
77
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1. Name the vertex and sides, then classify �IFT.(Lesson 1-4)
For Questions 2 and 3, complete the following proof. (Lesson 2-7)
Given: J�K� � L�M�H�J� � K�L�
Prove: H�K� � K�M�Proof:Statements Reasons1. J�K� � L�M�, H�J� � K�L� 1. Given2. JK � LM, HJ � KL 2.3. 3. Segment Addition Post.4. HJ � JK � KL � LM 4. Substitution Prop.5. HK � KM 5. Substitution Prop.6. H�K� � K�M� 6. Def. of � segments
For Questions 4 and 5, use the figure at the right.
4. Find the measure of the numbered angles if m�ABC � 57 and m�BCE � 98. (Lesson 4-2)
5. If B�D� is a median, AD � 2x � 6, and DC � 22.5 � 4x, find AC. (Lesson 5-1)
6. Write an inequality to describe the possible values of x. (Lesson 5-5)
7. A band of sequins that measures 108 inches is cut into twopieces so that their lengths are in a 5:7 ratio. Find the lengthof each piece. (Lesson 6-1)
8. Stan invests $1875 in a certificate of deposit that earns 4.5%interest compounded annually. Find the balance of his accountafter 4 years. (Lesson 6-6)
For Questions 9 and 10, use the figure at the right.
9. Find QP to the nearest tenth.(Lesson 7-2)
10. Find LM and PM. (Lesson 7-3)
60�30�10
9
P
ML
Q
R
5
85�117�
1212
145x � 6
E
CB
AD
25�1
23
4
5
41�
(Question 3)(Question 2)
H
J
K L M
F
TI 89�
NAME DATE PERIOD
SCORE
Standardized Test Practice (Chapters 1–7)
© Glencoe/McGraw-Hill 411 Glencoe Geometry
1. If T�A� bisects �YTB, T�C� bisects �BTZ,m�YTA � 4y � 6, and m�BTC � 7y � 4,find m�CTZ. (Lesson 1-4)
A. 52 B. 38C. 25 D. 8
2. Which statement is always true? (Lesson 2-5)
E. If right �QPR has sides q, p, and r, where r is the hypotenuse,then r2 � p2 � q2.
F. If E�F� || H�J�, then EF � HJ.G. If lines KL and VT are cut by a transversal, then K�L� || V�T�.H. If D�R� and R�H� are congruent, then R bisects D�H�.
3. The equation for PT��� is y � 2 � 8(x � 3). Determine an equationfor a line perpendicular to PT���. (Lesson 3-4)
A. y � �18�x � 7 B. y � 8x � 13
C. y � ��18�x � 2 D. y � �8x
4. Angle Y in �XYZ measures 90°. X�Y� and Y�Z� each measure 16meters. Classify �XYZ. (Lesson 4-1)
E. acute and isoscelesF. equiangular and equilateralG. right and scaleneH. right and isosceles
5. Two sides of a triangle measure 4 inches and 9 inches. Determinewhich cannot be the perimeter of the triangle. (Lesson 5-4)
A. 19 in. B. 21 in. C. 23 in. D. 26 in.
6. �ABC � �STR, so �AC
BA� � . (Lesson 6-2)
E. �BAB
C� F. �RST
S� G. �TR
RS� H. �
RST
S�
7. The Petronas Towers in Kuala Lumpur, Malaysia, are 452 meterstall. A woman who is 1.75 meters tall stands 120 meters from thebase of one tower. Find the angle of elevation between the woman’shat and the top of the tower to the nearest tenth. (Lesson 7-5)
A. 14.8° B. 15.4° C. 74.5° D. 75.1°
8. Which equation can be used to find x? (Lesson 7-4)
E. x � y sin 73° F. x � y cos 73°G. x � �cos
y73°� H. x � �sin
y73°�
73�
x
y
?
Y ZT
A CB
NAME DATE PERIOD
SCORE 77
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7.
8. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
© Glencoe/McGraw-Hill 412 Glencoe Geometry
Standardized Test Practice (continued)
9. Find the measure of the smaller of twocomplementary angles whose measures differby 23. (Lesson 1-5)
10. How many counterexamples are necessary toprove that a statement is false? (Lesson 2-3)
11. Find x so that � || m . (Lesson 3-5)
12. Find c to the nearest tenth. (Lesson 7-6)
26.1�
27.7�
126.2�
19
33
A
B
C
c
134 � 4x6x � 17
m�
NAME DATE PERIOD
77
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
9. 10.
11. 12.
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.
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13. If �DEF � �HJK, m�D � 26, m�J � 3x � 5, and m�F � 92, find x. (Lesson 4-3)
14. Use the Exterior Angle Inequality Theorem to list all of the angles whose measures are less than m�1. (Lesson 5-2)
For Questions 15 and 16, use the figure at the right.
15. Determine whether �EFH � �JGH. (Lesson 6-3)
16. If G is the midpoint of F�H�,find x. (Lesson 7-3)
60�18
30�E JH
F
Gx
1 2 4
3 56 7
8A DC
B
13.
14.
15.
16.
3 3 . 5 1
1 4 . 5 1 8 . 0
Unit 2 Review(Chapter 4–7)
77
© Glencoe/McGraw-Hill 413 Glencoe Geometry
Ass
essm
ents
1. Use a protractor to classify �UVW, �UWX,and �XWY as acute, equiangular, obtuse,or right.
2. In the figure, �1 � �2. Find the measures of the numbered angles.
3. Name the corresponding congruent sides for �AFP � �STX.
4. Determine whether �ABC � �PQR given A(2, �7), B(5, 3),C(�4, 6), P(8, �1), Q(11, 9), and R(2, 12).
5. In the figure, L�K� bisects �JKM and �KLJ � �KLM. Determine which theorem or postulate can be used to prove that �JKL � �MKL.
6. Triangle ABC is isosceles with AB � BC. Name a pair ofcongruent angles in this triangle.
7. Name the missing coordinates for isosceles right �JKL with legs b units long.
For Questions 8 and 9, refer to the figure.
8. Find a and m�ZWT if Z�W� is an altitude of �XYZ, m�ZWT � 3a � 5, and m�TWY � 5a � 13.
9. Determine which angle has the greatest measure: �YWZ,�WZY, or �ZYW.
10. Mr. Ramirez bought a stove and a dishwasher for just over$1206. State the assumption you would make to start anindirect proof to show that at least one of the appliances costmore than $603.
X W Y
T
Z
x
y
L(?, ?)
K(?, ?)
J(?, ?)
K L
J
M
65�
70�110�13
2
D
E F H
G
V
U
W
X
Y1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 414 Glencoe Geometry
Unit 2 Review (continued)77
11. Determine whether 128 feet, 136 feet, and 245 feet can be thelengths of the sides of a triangle.
12. Casey has a 13-inch television and a 52-inch television in herhome. What is the ratio of the sizes of the smaller and largerTVs?
13. If �EFG � �EJK, find x, JK,KG, and the scale factor relating �EFG to �EJK.
14. Find y.
15. Find the perimeter of �ABCif �ABC � �XYZ.
16. Alex has $750 in a bank account that earns 2.7% interest. Ifthe interest is compounded annually and he does not makeany withdrawals, find the balance of his account after 3 years.
17. Find the geometric mean between 27 and 42 to the nearesttenth.
18. Determine whether 27, 120, and 123 are the measures of thesides of a right triangle. Then state whether they form aPythagorean triple.
19. The diagonal of a square is 56 centimeters long. Find theperimeter of the square to the nearest tenth.
20. Find m�P to the nearest tenth in right �MNP for M(3, 6),N(3, �8), and P(�5, �8).
For Questions 21 and 22, refer to the figure.
21. Find m�S if m�T � 68, t � 65, and s � 33.
22. Solve �RST if t � 17, s � 11, and m�R � 40.
S
TR
r
s
t
X Z
Y
AC
B6
46
3028
y � 4
2y � 1
7
7
32�12
10
189
E
F
GK
J(x � 7)�
NAME DATE PERIOD
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
First Semester Test(Chapter 1–7)
77
© Glencoe/McGraw-Hill 415 Glencoe Geometry
Ass
essm
ents
For Questions 1–7, write the letter for the correct answer in the blank atthe right of each question.
1. Angles AFH and HFB form a linear pair and m�AFH � 83. Find m�HFB.A. 164 B. 97 C. 83 D. 41.5
2. Given C(2, 5), D(7, 0), and F(13, �6), which of the following is a trueconjecture?A. �CDF is a right triangle. B. �CDF is an isosceles triangle.C. �CDF is an equilateral triangle. D. C, D, and F do not form a triangle.
3. Which is the inverse of the statement If x � 5, then x � 3 � 8?A. If x � 3 � 8, then x � 5. B. If x � 5, then x � 3 � 8.C. If x � 5, then x � 3 � 8. D. If x � 3 � 8, then x � 5.
4. Find the slope of a line that is perpendicular to GH���.
A. �23� B. �
32�
C. � �23� D. � �
32�
5. Find the distance between parallel lines � and m whose equations are
y � �34�x � 4 and y � �
34�x � �
94�.
A. 4 B. 5 C. 9 D. �94�
6. Find sin P.
A. �15
40� B. �
14
48�
C. �45
80� D. 1
7. Find m�G.A. 30° B. 32°C. 35° D. 55.8°
32
2218
H
GF
50
48
14
P R
Q
x
y
O
G
H
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
6.
7.
8. Find c and PK if P is between L and K, LP � c � 22, PK � 5c,and LK � 34. Does P bisect L�K�?
9. Determine the distance between A(15, �12), and B(�30, 48) ona coordinate plane. State the coordinates of the midpoint of A�B�.
Justify each statement with a property or definition.
10. If A�C� � B�D�, then AC � BD.
11. If �2 and �3 are complementary, then m�2 � m�3 � 90.
© Glencoe/McGraw-Hill 416 Glencoe Geometry
First Semester Test (continued)77
12. If the measures of two angles of a triangle are 24 and 30, isthe triangle acute, obtuse, or right? Explain your reasoning.
13. Identify the congruent triangles in the figure.
For Questions 14 and 15, refer to the figure. Triangle ABC is anisosceles right triangle.
14. If C�D� bisects �C, find m�1 and m�2.
15. Determine the coordinates of A, B, and C, if the triangle haslegs n units long.
For Questions 16–18, refer to the figure.
16. Write a statement using �, �,or � to describe the measures of �DBC and �DCB.
17. Write an inequality to represent the possible measures of D�E�.
18. If m�FBC � 3x � 1 and m�CBD � 34, write an inequality todescribe the possible values of x.
19. Identify the similar triangles, find MN, and state the scale factor fromthe smaller triangle to the largertriangle.
20. Find the first three iterates of 4(x � 3) if x initially equals 0.
21. A plane is flying at 35,000 feet, and the pilot wants to descendto 22,000 feet over the next 60 miles. What should be his angleof depression to the nearest tenth? (Hint: 5280 feet � 1 mile)
22. Solve �DEF if DE � 58, EF � 62, and m�E � 49. Roundangle measures to the nearest degree and side measures tothe nearest tenth.
L
N
MK
J
5542
63
14
14
10
1213
F
B D
EC
x
y
C
B
A
D
1
2
K P
N
M
L
J
NAME DATE PERIOD
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
Standardized Test PracticeStudent Record Sheet (Use with pages 398–399 of the Student Edition.)
77
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5
3 6 DCBADCBA
DCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Questions 8, 9, 11, and 12, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.
8 (grid in) 8 9
9 (grid in)
10
11 (grid in)
12 (grid in)
11 12
0 0 0
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.
99 9 987654321
87654321
87654321
87654321
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.
99 9 987654321
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.
99 9 987654321
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.
99 9 987654321
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Record your answers for Question 13 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Geo
met
ric
Mea
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill35
1G
lenc
oe G
eom
etry
Lesson 7-1
Geo
met
ric
Mea
nT
he
geom
etri
c m
ean
betw
een
tw
o n
um
bers
is
the
squ
are
root
of
thei
r pr
odu
ct.F
or t
wo
posi
tive
nu
mbe
rs a
and
b,th
e ge
omet
ric
mea
n o
f a
and
bis
th
e po
siti
ve n
um
ber
xin
th
e pr
opor
tion
�a x��
� bx � .C
ross
mu
ltip
lyin
g gi
ves
x2�
ab,s
o x
��
ab�.
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mb
ers.
a.12
an
d 3
Let
xre
pres
ent
the
geom
etri
c m
ean
.
�1 x2 ��
� 3x �D
efin
ition
of
geom
etric
mea
n
x2�
36C
ross
mul
tiply
.
x�
�36�
or 6
Take
the
squ
are
root
of
each
sid
e.
b.
8 an
d 4
Let
xre
pres
ent
the
geom
etri
c m
ean
.
�8 x��
� 4x �
x2�
32x
��
32��
5.7
Exer
cises
Exer
cises
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mb
ers.
1.4
and
44
2.4
and
6�
24��
4.9
3.6
and
9�
54��
7.3
4.�1 2�
and
21
5.2�
3�an
d 3�
3��
18��
4.2
6.4
and
2510
7.�
3�an
d �
6�18
�1 4��
2.1
8.10
an
d 10
0�
1000
��
31.6
9.�1 2�
and
�1 4����1 8�
�0.
410
.an
d ���1 22 5�
�0.
7
11.4
an
d 16
812
.3 a
nd
24�
72��
8.5
Th
e ge
omet
ric
mea
n a
nd
on
e ex
trem
e ar
e gi
ven
.Fin
d t
he
oth
er e
xtre
me.
13.�
24�is
th
e ge
omet
ric
mea
n b
etw
een
aan
d b.
Fin
d b
if a
�2.
12
14.�
12�is
th
e ge
omet
ric
mea
n b
etw
een
aan
d b.
Fin
d b
if a
�3.
4
Det
erm
ine
wh
eth
er e
ach
sta
tem
ent
is a
lwa
ys,s
omet
imes
,or
nev
ertr
ue.
15.T
he
geom
etri
c m
ean
of
two
posi
tive
nu
mbe
rs i
s gr
eate
r th
an t
he
aver
age
of t
he
two
nu
mbe
rs.
nev
er
16.I
f th
e ge
omet
ric
mea
n o
f tw
o po
siti
ve n
um
bers
is
less
th
an 1
,th
en b
oth
of
the
nu
mbe
rsar
e le
ss t
han
1.
som
etim
es
3�2�
�5
2�2�
�5
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill35
2G
lenc
oe G
eom
etry
Alt
itu
de
of
a Tr
ian
gle
In t
he
diag
ram
,�A
BC
��
AD
B�
�B
DC
.A
n a
ltit
ude
to
the
hyp
oten
use
of
a ri
ght
tria
ngl
e fo
rms
two
righ
t tr
ian
gles
.Th
e tw
o tr
ian
gles
are
sim
ilar
an
d ea
ch i
s si
mil
ar t
o th
e or
igin
al t
rian
gle.
CDB
A
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Geo
met
ric
Mea
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
Use
rig
ht
�A
BC
wit
h
B�D�
⊥A�
C�.D
escr
ibe
two
geom
etri
cm
ean
s.
a.�
AD
B�
�B
DC
so � BA
D D��
�B CD D�
.
In �
AB
C,t
he
alti
tude
is
the
geom
etri
cm
ean
bet
wee
n t
he
two
segm
ents
of
the
hyp
oten
use
.
b.
�A
BC
��
AD
Ban
d �
AB
C�
�B
DC
,
so �A A
C B��
� AADB �
and
� BAC C�
�� DB
C C�.
In �
AB
C,e
ach
leg
is
the
geom
etri
cm
ean
bet
wee
n t
he
hyp
oten
use
an
d th
ese
gmen
t of
th
e h
ypot
enu
se a
djac
ent
toth
at l
eg.
Fin
d x
,y,a
nd
z.
� PPQR �
��P P
Q S�
�2 15 5��
�1 x5 �P
R�
25,
PQ
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, P
S�
x
25x
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tiply
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ivid
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ide
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5.
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en y�
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9�
16
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RS
�y
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� 1z 6�y
�16
z2�
400
Cro
ss m
ultip
ly.
z�
20Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
z
y
x
15
R QP
S25
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
,y,a
nd
zto
th
e n
eare
st t
enth
.
1.2.
3.
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7�
10��
3.2;
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7;3;
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35��
5.9
�8�
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8
4.5.
6.
2;3
2;�
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9
xz
y
62
x
zy
2
2x
y
1
��3
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zx
y 81
z
xy
5
2
x
13
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Geo
met
ric
Mea
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill35
3G
lenc
oe G
eom
etry
Lesson 7-1
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mb
ers.
Sta
te e
xact
an
swer
s an
dan
swer
s to
th
e n
eare
st t
enth
.
1.2
and
82.
9 an
d 36
3.4
and
7
418
�28�
�5.
3
4.5
and
105.
2�2�
and
5�2�
6.3�
5�an
d 5�
5�
�50�
�7.
1�
20��
4.5
�75�
�8.
7
Fin
d t
he
mea
sure
of
each
alt
itu
de.
Sta
te e
xact
an
swer
s an
d a
nsw
ers
to t
he
nea
rest
ten
th.
7.8.
�14�
�3.
7�
24��
4.9
9.10
.
�18�
�4.
26
Fin
d x
and
y.
11.
12.
6;�
108
��
10.4
�40�
�6.
3;�
56��
7.5
13.
14.
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7;�
285
��
16.9
12.5
;�
29��
5.425y
x
15
4
y
x
10
4
yx
39y
x
RT
S U4.
58
GEH
F
2
9
L
M N
P2
12
C
D
B
A2
7
©G
lenc
oe/M
cGra
w-H
ill35
4G
lenc
oe G
eom
etry
Fin
d t
he
geom
etri
c m
ean
bet
wee
n e
ach
pai
r of
nu
mb
ers
to t
he
nea
rest
ten
th.
1.8
and
122.
3�7�
and
6�7�
3.�4 5�
and
2
�96�
�9.
8�
126
��
11.2
���8 5��
1.3
Fin
d t
he
mea
sure
of
each
alt
itu
de.
Sta
te e
xact
an
swer
s an
d a
nsw
ers
to t
he
nea
rest
ten
th.
4.5.
�60�
�7.
7�
102
��
10.1
Fin
d x
,y,a
nd
z.
6.7.
�18
4�
�13
.6;
�24
8�
�15
.7;
�11
4�
�10
.7;
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0�
�12
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3�
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21.8
8.9.
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515
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.3
10.C
IVIL
EN
GIN
EER
ING
An
airp
ort,
a fa
ctor
y,an
d a
shop
ping
cen
ter
are
at t
he v
erti
ces
of a
righ
t tr
iang
le f
orm
ed b
y th
ree
high
way
s.T
he a
irpo
rt a
nd f
acto
ry a
re 6
.0 m
iles
apa
rt.T
heir
dist
ance
s fr
om t
he s
hopp
ing
cent
er a
re 3
.6 m
iles
and
4.8
mil
es,r
espe
ctiv
ely.
A s
ervi
ce r
oad
wil
l be
con
stru
cted
fro
m t
he s
hopp
ing
cent
er t
o th
e hi
ghw
ay t
hat
conn
ects
the
air
port
and
fact
ory.
Wh
at i
s th
e sh
orte
st p
ossi
ble
len
gth
for
th
e se
rvic
e ro
ad?
Rou
nd
to t
he
nea
rest
hu
ndr
edth
.2.
88 m
i
xy10
z
20x
y 2
3
z
zx
y625
23
z
xy
8
17
6
KLJ
M
125
U
TA
V
Pra
ctic
e (
Ave
rag
e)
Geo
met
ric
Mea
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csG
eom
etri
c M
ean
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill35
5G
lenc
oe G
eom
etry
Lesson 7-1
Pre-
Act
ivit
yH
ow c
an t
he
geom
etri
c m
ean
be
use
d t
o vi
ew p
ain
tin
gs?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-1
at
the
top
of p
age
342
in y
our
text
book
.
•W
hat
is
a di
sadv
anta
ge o
f st
andi
ng
too
clos
e to
a p
ain
tin
g?S
amp
le a
nsw
er:Y
ou
do
n’t
get
a g
oo
d o
vera
ll vi
ew.
•W
hat
is
a di
sadv
anta
ge o
f st
andi
ng
too
far
from
a p
ain
tin
g?S
amp
le a
nsw
er:Y
ou
can
’t s
ee a
ll th
e d
etai
ls in
th
e p
ain
tin
g.
Rea
din
g t
he
Less
on
1.In
th
e pa
st,w
hen
you
hav
e se
en t
he
wor
d m
ean
in m
ath
emat
ics,
it r
efer
red
to t
he
aver
age
or a
rith
met
ic m
ean
of t
he
two
nu
mbe
rs.
a.C
ompl
ete
the
foll
owin
g by
wri
tin
g an
alg
ebra
ic e
xpre
ssio
n i
n e
ach
bla
nk.
If a
and
bar
e tw
o po
siti
ve n
um
bers
,th
en t
he
geom
etri
c m
ean
bet
wee
n a
and
bis
and
thei
r ar
ith
met
ic m
ean
is
.
b.
Exp
lain
in
wor
ds,w
ith
out
usi
ng
any
mat
hem
atic
al s
ymbo
ls,t
he
diff
eren
ce b
etw
een
the
geom
etri
c m
ean
an
d th
e al
gebr
aic
mea
n.
Sam
ple
an
swer
:Th
e g
eom
etri
cm
ean
bet
wee
n t
wo
nu
mb
ers
is t
he
squ
are
roo
t o
f th
eir
pro
du
ct.T
he
arit
hm
etic
mea
n o
f tw
o n
um
ber
s is
hal
f th
eir
sum
.
2.L
et r
and
sbe
tw
o po
siti
ve n
um
bers
.In
wh
ich
of
the
foll
owin
g eq
uat
ion
s is
zeq
ual
to
the
geom
etri
c m
ean
bet
wee
n r
and
s?A
,C,D
,FA
.� zs �
��z r�
B.
� zr ��
� zs �C
.s:
z�
z:r
D.
� zr ��
�z s�E
.�z r�
��z s�
F.�z s�
�� zr �
3.S
upp
ly t
he
mis
sin
g w
ords
or
phra
ses
to c
ompl
ete
the
stat
emen
t of
eac
h t
heo
rem
.
a.T
he m
easu
re o
f th
e al
titu
de d
raw
n fr
om t
he v
erte
x of
the
rig
ht a
ngle
of
a ri
ght
tria
ngle
to i
ts h
ypot
enu
se i
s th
e be
twee
n t
he
mea
sure
s of
th
e tw
o
segm
ents
of
the
.
b.
If t
he
alti
tude
is
draw
n f
rom
th
e ve
rtex
of
the
angl
e of
a r
igh
t
tria
ngl
e to
its
hyp
oten
use
,th
en t
he
mea
sure
of
a of
th
e tr
ian
gle
is t
he
betw
een
the
mea
sure
of
the
hypo
tenu
se a
nd t
he s
egm
ent
of t
he
adja
cen
t to
th
at l
eg.
c.If
th
e al
titu
de i
s dr
awn
fro
m t
he
of t
he
righ
t an
gle
of a
rig
ht
tria
ngl
e to
its
,t
hen
th
e tw
o tr
ian
gles
for
med
are
to t
he
give
n t
rian
gle
and
to e
ach
oth
er.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al c
once
pt i
s to
rel
ate
it t
o so
met
hin
g yo
ual
read
y kn
ow.H
ow c
an t
he
mea
nin
g of
mea
nin
a p
ropo
rtio
n h
elp
you
to
rem
embe
r h
owto
fin
d th
e ge
omet
ric
mea
n b
etw
een
tw
o n
um
bers
?S
amp
le a
nsw
er:W
rite
ap
rop
ort
ion
in w
hic
h t
he
two
mea
ns
are
equ
al.T
his
co
mm
on
mea
n is
th
eg
eom
etri
c m
ean
bet
wee
n t
he
two
ext
rem
es.
sim
ilar
hyp
ote
nu
seve
rtex
hyp
ote
nu
seg
eom
etri
c m
ean
leg
rig
ht
hyp
ote
nu
seg
eom
etri
c m
ean
�a� 2
b�
�ab�
©G
lenc
oe/M
cGra
w-H
ill35
6G
lenc
oe G
eom
etry
Mat
hem
atic
s an
d M
usi
cP
yth
agor
as,a
Gre
ek p
hil
osop
her
wh
o li
ved
duri
ng
the
sixt
h c
entu
ry B
.C.,
beli
eved
th
at a
ll n
atu
re,b
eau
ty,a
nd
har
mon
y co
uld
be
expr
esse
d by
who
le-
num
ber
rela
tion
ship
s.M
ost
peop
le r
emem
ber
Pyt
hag
oras
for
his
tea
chin
gsab
out
righ
t tr
ian
gles
.(T
he
sum
of
the
squ
ares
of
the
legs
equ
als
the
squ
are
ofth
e h
ypot
enu
se.)
Bu
t P
yth
agor
as a
lso
disc
over
ed r
elat
ion
ship
s be
twee
n t
he
mu
sica
l n
otes
of
a sc
ale.
Th
ese
rela
tion
ship
s ca
n b
e ex
pres
sed
as r
atio
s.
CD
EF
GA
BC
�
�1 1��8 9�
�4 5��3 4�
�2 3��3 5�
� 18 5��1 2�
Wh
en y
ou p
lay
a st
rin
ged
inst
rum
ent,
Th
e C
str
ing
can
be
use
dyo
u p
rodu
ce d
iffe
ren
t n
otes
by
plac
ing
to p
rodu
ce F
by
plac
ing
you
r fi
nge
r on
dif
fere
nt
plac
es o
n a
str
ing.
a fi
nge
r �3 4�
of t
he
way
Th
is i
s th
e re
sult
of
chan
gin
g th
e le
ngt
hal
ong
the
stri
ng.
of t
he
vibr
atin
g pa
rt o
f th
e st
rin
g.
Su
pp
ose
a C
str
ing
has
a l
engt
h o
f 16
in
ches
.Wri
te a
nd
sol
ve
pro
por
tion
s to
det
erm
ine
wh
at l
engt
h o
f st
rin
g w
ould
hav
e to
vi
bra
te t
o p
rod
uce
th
e re
mai
nin
g n
otes
of
the
scal
e.
1.D
14�2 9�
in.
2.E
12�4 5�
in.
3.F
12 in
.
4.G
10�2 3�
in.
5.A
9�3 5�
in.
6.B
8� 18 5�
in.
7.C
�8
in.
8.C
ompl
ete
to s
how
th
e di
stan
ce b
etw
een
fin
ger
posi
tion
s on
th
e 16
-in
ch
C s
trin
g fo
r ea
ch n
ote.
For
exa
mpl
e,C
(16)
�D
�14�2 9� �
�1�
7 9� .
C
D
E
F
G
A
B
C�
9.B
etw
een
tw
o co
nse
cuti
ve m
usi
cal
not
es,t
her
e is
eit
her
a w
hol
e st
ep o
r a
hal
f st
ep.U
sin
g th
e di
stan
ces
you
fou
nd
in E
xerc
ise
8,de
term
ine
wh
at
two
pair
s of
not
es h
ave
a h
alf
step
bet
wee
n t
hem
.
Ean
d F
,Ban
d C
�
8 in
.7�
17 5�in
.6�
2 5�in
.5�
1 3�in
.4
in.
3�1 5�
in.
1�7 9�
in.En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-1
7-1
3 4of
C s
tring
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Th
e P
yth
ago
rean
Th
eore
m a
nd
Its
Co
nver
se
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill35
7G
lenc
oe G
eom
etry
Lesson 7-2
The
Pyth
ago
rean
Th
eore
mIn
a r
igh
t tr
ian
gle,
the
sum
of
the
sq
uar
es o
f th
e m
easu
res
of t
he
legs
equ
als
the
squ
are
of t
he
mea
sure
of
the
hyp
oten
use
.�
AB
Cis
a r
igh
t tr
ian
gle,
so a
2�
b2�
c2.
Pro
ve t
he
Pyt
hag
orea
n T
heo
rem
.W
ith
alt
itu
de C�
D�,e
ach
leg
aan
d b
is a
geo
met
ric
mea
n b
etw
een
h
ypot
enu
se c
and
the
segm
ent
of t
he
hyp
oten
use
adj
acen
t to
th
at l
eg.
� ac ��
�a y�an
d � bc �
��b x� ,
so a
2�
cyan
d b2
�cx
.
Add
th
e tw
o eq
uat
ion
s an
d su
bsti
tute
c�
y�
xto
get
a2�
b2�
cy�
cx�
c(y
�x)
�c2
.
cy
xa
b
hA
CBD
ca
bA
CB
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
a.F
ind
a.
a2�
b2�
c2P
ytha
gore
an T
heor
em
a2�
122
�13
2b
�12
, c
�13
a2�
144
�16
9S
impl
ify.
a2�
25S
ubtr
act.
a�
5Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
a
1213
ACB
b.
Fin
d c
.
a2�
b2�
c2P
ytha
gore
an T
heor
em
202
�30
2�
c2a
�20
, b
�30
400
�90
0 �
c2S
impl
ify.
1300
�c2
Add
.
�13
00�
�c
Take
the
squ
are
root
of
each
sid
e.
36.1
�c
Use
a c
alcu
lato
r.
c
30
20
ACB
Exer
cises
Exer
cises
Fin
d x
.
1.2.
3.
�18�
�4.
212
60
4.5.
6.
� 13 0��
1345
��
36.7
�66
3�
�25
.7x
1128
x
33
16x
5 9
4 9
x6525
x 159
x
33
©G
lenc
oe/M
cGra
w-H
ill35
8G
lenc
oe G
eom
etry
Co
nve
rse
of
the
Pyth
ago
rean
Th
eore
mIf
th
e su
m o
f th
e sq
uar
es
of t
he
mea
sure
s of
th
e tw
o sh
orte
r si
des
of a
tri
angl
e eq
ual
s th
e sq
uar
e of
th
e m
easu
re o
f th
e lo
nge
st s
ide,
then
th
e tr
ian
gle
is a
rig
ht
tria
ngl
e.
If t
he
thre
e w
hol
e n
um
bers
a,b
,an
d c
sati
sfy
the
equ
atio
n
a2�
b2�
c2,t
hen
th
e n
um
bers
a,b
,an
d c
form
a
If a2
�b2
�c2
, th
en
Pyt
hag
orea
n t
rip
le.
�A
BC
is a
rig
ht t
riang
le.
Det
erm
ine
wh
eth
er �
PQ
Ris
a r
igh
t tr
ian
gle.
a2�
b2�
c2P
ytha
gore
an T
heor
em
102
�(1
0�3�)
2�
202
a�
10,
b�
10�
3�, c
�20
100
�30
0 �
400
Sim
plify
.
400
�40
0✓A
dd.
Th
e su
m o
f th
e sq
uar
es o
f th
e tw
o sh
orte
r si
des
equ
als
the
squ
are
of t
he
lon
gest
sid
e,so
th
etr
ian
gle
is a
rig
ht
tria
ngl
e.
Det
erm
ine
wh
eth
er e
ach
set
of
mea
sure
s ca
n b
e th
e m
easu
res
of t
he
sid
es o
f a
righ
t tr
ian
gle.
Th
en s
tate
wh
eth
er t
hey
for
m a
Pyt
hag
orea
n t
rip
le.
1.30
,40,
502.
20,3
0,40
3.18
,24,
30
yes;
yes
no
;n
oye
s;ye
s
4.6,
8,9
5.�3 7� ,
�4 7� ,�5 7�
6.10
,15,
20
no
;n
oye
s;n
on
o;
no
7.�
5�,�
12�,�
13�8.
2,�
8�,�
12�9.
9,40
,41
no
;n
oye
s;n
oye
s;ye
s
A f
am
ily
of P
yth
agor
ean
tri
ple
s co
nsi
sts
of m
ult
iple
s of
kn
own
tri
ple
s.F
or e
ach
Pyt
hag
orea
n t
rip
le,f
ind
tw
o tr
iple
s in
th
e sa
me
fam
ily.
Sam
ple
an
swer
s ar
eg
iven
.10
.3,4
,511
.5,1
2,13
12.7
,24,
25
30,4
0,50
;10
,24,
26;
14,4
8,50
;12
,16,
2015
,36,
3921
,72,
75
10�
�3
2010 Q
R
P
c
ab
A
C
B
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Th
e P
yth
ago
rean
Th
eore
m a
nd
Its
Co
nver
se
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
Th
e P
yth
ago
rean
Th
eore
m a
nd
Its
Co
nver
se
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill35
9G
lenc
oe G
eom
etry
Lesson 7-2
Fin
d x
.
1.2.
3.
155
�11
68�
�34
.2
4.5.
6.
�46
8.7
�5�
�21
.7�
65��
8.1
�11
57�
�34
.0
Det
erm
ine
wh
eth
er �
ST
Uis
a r
igh
t tr
ian
gle
for
the
give
n v
erti
ces.
Exp
lain
.
7.S
(5,5
),T
(7,3
),U
(3,2
)8.
S(3
,3),
T(5
,5),
U(6
,0)
no
;S
T�
�8�,
TU
��
17�,
yes;
ST
��
8�,T
U�
�26�
,
US
��
13�,
US
��
18�,
( �8�)
2�
( �13�
)2�
( �17�
)2( �
8�)2
� ( �
18�)2
�( �
26�)2
9.S
(4,6
),T
(9,1
),U
(1,3
)10
.S(0
,3),
T(�
2,5)
,U(4
,7)
yes;
ST
��
50�,T
U�
�68�
,ye
s;S
T�
�8�,
TU
��
40�,
US
��
18�,
US
��
32�,
( �18�
)2�
( �50�
)2�
( �68�
)2( �
8�)2
� ( �
32�)2
�( �
40�)2
11.S
(�3,
2),T
(2,7
),U
(�1,
1)12
.S(2
,�1)
,T(5
,4),
U(6
,�3)
yes;
ST
��
50�,T
U�
�45�
,n
o;
ST
��
34�,T
U�
�50�
,
US
��
5�,U
S�
�20�
,( �
45�)2
� ( �
5�)2
�( �
50�)2
( �34�
)2�
( �20�
)2�
( �50�
)2
Det
erm
ine
wh
eth
er e
ach
set
of
mea
sure
s ca
n b
e th
e m
easu
res
of t
he
sid
es o
f a
righ
t tr
ian
gle.
Th
en s
tate
wh
eth
er t
hey
for
m a
Pyt
hag
orea
n t
rip
le.
13.1
2,16
,20
14.1
6,30
,32
15.1
4,48
,50
yes,
yes
no
,no
yes,
yes
16.�
2 5� ,�4 5� ,
�6 5�17
.2�
6�,5,
718
.2�
2�,2�
7�,6
no
,no
yes,
no
yes,
nox31
14x
99
8
x12
.5
25
x
1232
x
12
13x 12
9
©G
lenc
oe/M
cGra
w-H
ill36
0G
lenc
oe G
eom
etry
Fin
d x
.
1.2.
3.
�69
8�
�26
.4�
715
��
26.7
�59
5�
�24
.4
4.5.
6.
�16
40�
�40
.5�
60��
7.7
�13
5�
�11
.6
Det
erm
ine
wh
eth
er �
GH
Iis
a r
igh
t tr
ian
gle
for
the
give
n v
erti
ces.
Exp
lain
.
7.G
(2,7
),H
(3,6
),I(
�4,
�1)
8.G
(�6,
2),H
(1,1
2),I
(�2,
1)
yes;
GH
��
2�,H
I��
98�,
no
;G
H�
�14
9�
,HI�
�13
0�
,
IG�
�10
0�
,IG
��
17�,
( �2�)
2�
( �98�
)2�
( �10
0�
)2( �
130
�)2
� ( �
17�)2
�( �
149
�)2
9.G
(�2,
1),H
(3,�
1),I
(�4,
�4)
10.G
(�2,
4),H
(4,1
),I(
�1,
�9)
yes;
GH
��
29�,H
I��
58�,
yes;
GH
��
45�,H
I� �
125
�,
IG�
�29�
,IG
��
170
�,
( �29�
)2�
( �29�
)2�
( �58�
)2( �
45�)2
� ( �
125
�)2
�( �
170
�)2
Det
erm
ine
wh
eth
er e
ach
set
of
mea
sure
s ca
n b
e th
e m
easu
res
of t
he
sid
es o
f a
righ
t tr
ian
gle.
Th
en s
tate
wh
eth
er t
hey
for
m a
Pyt
hag
orea
n t
rip
le.
11.9
,40,
4112
.7,2
8,29
13.2
4,32
,40
yes,
yes
no
,no
yes,
yes
14.�
9 5� ,�1 52 �
,315
.�1 3� ,
,116
.,
,�4 7�
yes,
no
yes,
no
yes,
no
17.C
ON
STR
UC
TIO
NT
he
bott
om e
nd
of a
ram
p at
a w
areh
ouse
is
10 f
eet
from
th
e ba
se o
f th
e m
ain
doc
k an
d is
11
feet
lon
g.H
ow
hig
h i
s th
e do
ck?
abo
ut
4.6
ft h
igh
11 ft
?dock
ram
p
10 ft
2�3�
�7
�4�
�7
2 �2�
�3
x24
24
42
x16
14
x
34
22
x26
2618
x
3421
x
13
23Pra
ctic
e (
Ave
rag
e)
Th
e P
yth
ago
rean
Th
eore
m a
nd
Its
Co
nver
se
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csT
he
Pyt
hag
ore
an T
heo
rem
an
d It
s C
onv
erse
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill36
1G
lenc
oe G
eom
etry
Lesson 7-2
Pre-
Act
ivit
yH
ow a
re r
igh
t tr
ian
gles
use
d t
o b
uil
d s
usp
ensi
on b
rid
ges?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-2
at
the
top
of p
age
350
in y
our
text
book
.
Do
the
two
righ
t tr
ian
gles
sh
own
in
th
e dr
awin
g ap
pear
to
be s
imil
ar?
Exp
lain
you
r re
ason
ing.
Sam
ple
an
swer
:N
o;
thei
r si
des
are
no
tp
rop
ort
ion
al.I
n t
he
smal
ler
tria
ng
le,t
he
lon
ger
leg
is m
ore
th
antw
ice
the
len
gth
of
the
sho
rter
leg
,wh
ile in
th
e la
rger
tri
ang
le,
the
lon
ger
leg
is le
ss t
han
tw
ice
the
len
gth
of
the
sho
rter
leg
.
Rea
din
g t
he
Less
on
1.E
xpla
in i
n y
our
own
wor
ds t
he
diff
eren
ce b
etw
een
how
th
e P
yth
agor
ean
Th
eore
m i
s u
sed
and
how
th
e C
onve
rse
of t
he
Pyt
hag
orea
n T
heo
rem
is
use
d.S
amp
le a
nsw
er:T
he
Pyt
hag
ore
an T
heo
rem
is u
sed
to
fin
d t
he
thir
d s
ide
of
a ri
gh
t tr
ian
gle
ifyo
u k
no
w t
he
len
gth
s o
f an
y tw
o o
f th
e si
des
.Th
e co
nver
se is
use
d t
ote
ll w
het
her
a t
rian
gle
wit
h t
hre
e g
iven
sid
e le
ng
ths
is a
rig
ht
tria
ng
le.
2.R
efer
to
the
figu
re.F
or t
his
fig
ure
,wh
ich
sta
tem
ents
are
tru
e?
A.
m2
�n
2�
p2B
.n2
�m
2�
p2B
,E,F
,G
C.
m2
�n
2�
p2D
.m2
�p2
� n
2
E.
p2�
n2
� m
2F.
n2
� p
2�
m2
G.n
��
m2
��
p2 �H
.p�
�m
2�
�n
2 �
3.Is
th
e fo
llow
ing
stat
emen
t tr
ue
or f
alse
?A
Pyt
hago
rean
tri
ple
is a
ny g
roup
of
thre
e nu
mbe
rs f
or w
hich
the
sum
of
the
squa
res
of t
hesm
alle
r tw
o nu
mbe
rs is
equ
al t
o th
e sq
uare
of
the
larg
est
num
ber.
Exp
lain
you
r re
ason
ing.
Sam
ple
an
swer
:Th
e st
atem
ent
is f
alse
bec
ause
in a
Pyt
hag
ore
an t
rip
le,
all t
hre
e n
um
ber
s m
ust
be
wh
ole
nu
mb
ers.
4.If
x,y
,an
d z
form
a P
yth
agor
ean
tri
ple
and
kis
a p
osit
ive
inte
ger,
wh
ich
of
the
foll
owin
ggr
oups
of
nu
mbe
rs a
re a
lso
Pyt
hag
orea
n t
ripl
es?
B,D
A.3
x,4y
,5z
B.3
x,3y
,3z
C.�
3x,�
3y,�
3zD
.kx,
ky,k
z
Hel
pin
g Y
ou
Rem
emb
er
5.M
any
stud
ents
who
stu
died
geo
met
ry lo
ng a
go r
emem
ber
the
Pyt
hago
rean
The
orem
as
the
equa
tion
a2
�b2
�c2
,but
can
not
tell
you
wha
t th
is e
quat
ion
mea
ns.A
for
mul
a is
use
less
if y
ou d
on’t
know
wh
at i
t m
ean
s an
d h
ow t
o u
se i
t.H
ow c
ould
you
hel
p so
meo
ne
wh
o h
asfo
rgot
ten
the
Pyt
hago
rean
The
orem
rem
embe
r th
e m
eani
ng o
f th
e eq
uati
on a
2�
b2�
c2?
Sam
ple
an
swer
:D
raw
a r
igh
t tr
ian
gle
.Lab
el t
he
len
gth
s o
f th
e tw
o le
gs
as a
and
ban
d t
he
len
gth
of
the
hyp
ote
nu
se a
s c.
pm
n
©G
lenc
oe/M
cGra
w-H
ill36
2G
lenc
oe G
eom
etry
Co
nver
se o
f a
Rig
ht T
rian
gle
Th
eore
mYo
u h
ave
lear
ned
th
at t
he
mea
sure
of
the
alti
tude
fro
m t
he
vert
ex o
fth
e ri
ght
angl
e of
a r
igh
t tr
ian
gle
to i
ts h
ypot
enu
se i
s th
e ge
omet
ric
mea
n b
etw
een
th
e m
easu
res
of t
he
two
segm
ents
of
the
hyp
oten
use
.Is
th
e co
nve
rse
of t
his
th
eore
m t
rue?
In
ord
er t
o fi
nd
out,
it w
ill
hel
pto
rew
rite
th
e or
igin
al t
heo
rem
in
if-
then
for
m a
s fo
llow
s.
If �
AB
Qis
a r
igh
t tr
ian
gle
wit
h r
igh
t an
gle
at Q
,th
en
QP
is
the
geom
etri
c m
ean
bet
wee
n A
Pan
d P
B,w
her
e P
is b
etw
een
Aan
d B
and
Q �P �
is p
erpe
ndi
cula
r to
A �B �
.
1.W
rite
th
e co
nve
rse
of t
he
if-t
hen
for
m o
f th
e th
eore
m.
If Q
Pis
th
e g
eom
etri
c m
ean
bet
wee
n A
Pan
d
PB
,wh
ere
Pis
bet
wee
n A
and
Ban
d Q �
P ��
A �B �
,th
en �
AB
Qis
a r
igh
t tr
ian
gle
wit
h r
igh
t an
gle
at
Q.
2.Is
th
e co
nve
rse
of t
he
orig
inal
th
eore
m t
rue?
Ref
er
to t
he
figu
re a
t th
e ri
ght
to e
xpla
in y
our
answ
er.
Yes;
(PQ
)2�
(AP
)(P
B)
imp
lies
that
�P AQ P��
� PPQB �
.
Sin
ce b
oth
�A
PQ
and
�Q
PB
are
rig
ht
ang
les,
they
are
co
ng
ruen
t.T
her
efo
re�
AP
Q�
�Q
PB
by S
AS
sim
ilari
ty.S
o
�A
��
PQ
Ban
d �
AQ
P�
�B
.Bu
t th
e ac
ute
an
gle
s o
f �
AQ
Par
e co
mp
lem
enta
ry a
nd
m
�A
QB
�m
�A
QP
�m
�P
QB
.Hen
ce
m�
AQ
B�
90 a
nd
�A
QB
is a
rig
ht
tria
ng
le
wit
h r
igh
t an
gle
at
Q.
You
may
fin
d it
in
tere
stin
g to
exa
min
e th
e ot
her
th
eore
ms
inC
hap
ter
7 to
see
wh
eth
er t
hei
r co
nve
rses
are
tru
e or
fal
se.Y
ou w
ill
nee
d to
res
tate
th
e th
eore
ms
care
full
y in
ord
er t
o w
rite
th
eir
con
vers
es.
Q
BP
A
Q
BP
A
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-2
7-2
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sp
ecia
l Rig
ht T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill36
3G
lenc
oe G
eom
etry
Lesson 7-3
Pro
per
ties
of
45°-
45°-
90°
Tria
ng
les
Th
e si
des
of a
45°
-45°
-90°
righ
t tr
ian
gle
hav
e a
spec
ial
rela
tion
ship
.
If t
he
leg
of a
45°
-45°
-90°
righ
t tr
ian
gle
is x
un
its,
show
th
at t
he
hyp
oten
use
is
x�2�
un
its.
Usi
ng
the
Pyt
hag
orea
n T
heo
rem
wit
h
a�
b�
x,th
en
c2�
a2�
b2
�x2
�x2
�2x
2
c�
�2x
2�
�x�
2�
x��
x
x2
45�
45�
In a
45°
-45°
-90°
righ
t tr
ian
gle
the
hyp
oten
use
is
�2�
tim
es
the
leg.
If t
he
hyp
oten
use
is
6 u
nit
s,fi
nd
th
e le
ngt
h o
f ea
ch l
eg.
Th
e h
ypot
enu
se i
s �
2�ti
mes
th
e le
g,so
divi
de t
he
len
gth
of
the
hyp
oten
use
by
�2�.
a� � � �
3 �2�
un
its
6�2�
�26�
2�� �
2��2�
6� �
2�
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
.
1.2.
3.
8�2�
�11
.33
5�2�
�7.
1
4.5.
6.
9�2�
�12
.718
�2�
�25
.56
7.F
ind
the
peri
met
er o
f a
squ
are
wit
h d
iago
nal
12
cen
tim
eter
s.24
�2�
�33
.9 c
m
8.F
ind
the
diag
onal
of
a sq
uar
e w
ith
per
imet
er 2
0 in
ches
.5�
2��
7.1
in.
9.F
ind
the
diag
onal
of
a sq
uar
e w
ith
per
imet
er 2
8 m
eter
s.7�
2��
9.9
m
x3�
�2x
18x
x
18
x10
x
45�
3��2
x 8
45�
45�
©G
lenc
oe/M
cGra
w-H
ill36
4G
lenc
oe G
eom
etry
Pro
per
ties
of
30°-
60°-
90°
Tria
ng
les
Th
e si
des
of a
30°
-60°
-90°
righ
t tr
ian
gle
also
hav
e a
spec
ial
rela
tion
ship
.
In a
30°
-60°
-90°
righ
t tr
ian
gle,
show
th
at t
he
hyp
oten
use
is
twic
e th
e sh
orte
r le
g an
d t
he
lon
ger
leg
is �
3�ti
mes
th
e sh
orte
r le
g.
�M
NQ
is a
30°
-60°
-90°
righ
t tr
ian
gle,
and
the
len
gth
of
the
hyp
oten
use
M �N�
is t
wo
tim
es t
he
len
gth
of
the
shor
ter
side
N�Q�
.U
sin
g th
e P
yth
agor
ean
Th
eore
m,
a2�
(2x)
2�
x2
�4x
2�
x2
�3x
2
a�
�3x
2�
�x�
3�
In a
30°
-60°
-90°
righ
t tr
ian
gle,
the
hyp
oten
use
is
5 ce
nti
met
ers.
Fin
d t
he
len
gth
s of
th
e ot
her
tw
o si
des
of
the
tria
ngl
e.If
th
e h
ypot
enu
se o
f a
30°-
60°-
90°
righ
t tr
ian
gle
is 5
cen
tim
eter
s,th
en t
he
len
gth
of
the
shor
ter
leg
is h
alf
of 5
or
2.5
cen
tim
eter
s.T
he
len
gth
of
the
lon
ger
leg
is �
3�ti
mes
th
e le
ngt
h o
f th
e sh
orte
r le
g,or
(2.
5)(�
3�)ce
nti
met
ers.
Fin
d x
and
y.
1.2.
3.
1;0.
5�3�
�0.
98�
3��
13.9
;16
5.5;
5.5�
3��
9.5
4.5.
6.
9;18
4�3�
�6.
9;8�
3��
13.9
10�
3��
17.3
;10
7.T
he
peri
met
er o
f an
equ
ilat
eral
tri
angl
e is
32
cen
tim
eter
s.F
ind
the
len
gth
of
an a
ltit
ude
of t
he
tria
ngl
e to
th
e n
eare
st t
enth
of
a ce
nti
met
er.
9.2
cm
8.A
n a
ltit
ude
of
an e
quil
ater
al t
rian
gle
is 8
.3 m
eter
s.F
ind
the
peri
met
er o
f th
e tr
ian
gle
toth
e n
eare
st t
enth
of
a m
eter
.28
.8 m
xy
60�
20
xy
60�
12
xy
30�
9 ��3
x
y11
30�
x
y
60� 8
x y30
�
60�
1 2
x
a
NQP
M
2x30�
30�
60�
60�
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Sp
ecia
l Rig
ht T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
�M
NP
is a
n eq
uila
tera
ltr
iang
le.
�M
NQ
is a
30°
-60°
-90°
right
tria
ngle
.
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Sp
ecia
l Rig
ht T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill36
5G
lenc
oe G
eom
etry
Lesson 7-3
Fin
d x
and
y.
1.2.
3.
12,1
2�3�
64,3
2�3�
6�2�,
6�2�
4.5.
6.
8,8�
2�8,
8�3�
45,1
3�2�
For
Exe
rcis
es 7
–9,u
se t
he
figu
re a
t th
e ri
ght.
7.If
a�
11,f
ind
ban
d c.
b�
11�
3�;c
�22
8.If
b�
15,f
ind
aan
d c.
a�
5�3�;
c�
10�
3�
9.If
c�
9,fi
nd
aan
d b.
a�
4.5;
b�
4.5�
3�
For
Exe
rcis
es 1
0 an
d 1
1,u
se t
he
figu
re a
t th
e ri
ght.
10.T
he
peri
met
er o
f th
e sq
uar
e is
30
inch
es.F
ind
the
len
gth
of
B�C�
.
7.5
in.
11.F
ind
the
len
gth
of
the
diag
onal
B�D�
.
7.5�
2�in
.or
abo
ut
10.6
1 in
.
12.T
he
peri
met
er o
f th
e eq
uil
ater
al t
rian
gle
is 6
0 m
eter
s.F
ind
the
le
ngt
h o
f an
alt
itu
de.
10�
3�m
or
abo
ut
17.3
2 m
13.�
GE
Cis
a 3
0°-6
0°-9
0°tr
ian
gle
wit
h r
igh
t an
gle
at E
,an
d E�
C�is
th
e lo
nge
r le
g.F
ind
the
coor
din
ates
of
Gin
Qu
adra
nt
I fo
r E
(1,1
) an
d C
(4,1
).
( 1,1
��
3�)o
r ab
ou
t (1
,2.7
3)
E
FG
D60
�
AB C
D45
�
bA
B Cac
60�
30�
y x� 13
1313
13
y
x60�
16
y
x
45�
8
y
x
45�
12
y
x
30�
32
y
x60
�24
©G
lenc
oe/M
cGra
w-H
ill36
6G
lenc
oe G
eom
etry
Fin
d x
and
y.
1.2.
3.
,25
�3�,
5013
,13�
3�
4.5.
6.
45,1
4�2�
3.5�
3�,7
;11
�2�
For
Exe
rcis
es 7
–9,u
se t
he
figu
re a
t th
e ri
ght.
7.If
a�
4�3�,
fin
d b
and
c.
b�
12,c
�8�
3�
8.If
x�
3�3�,
fin
d a
and
CD
.
a�
6�3�,
CD
�9
9.If
a�
4,fi
nd
CD
,b,a
nd
y.
CD
�2�
3�,b
�4�
3�,y
�6
10.T
he
peri
met
er o
f an
equ
ilat
eral
tri
angl
e is
39
cen
tim
eter
s.F
ind
the
len
gth
of
an a
ltit
ude
of t
he
tria
ngl
e.
6.5�
3�in
.or
abo
ut
11.2
6 in
.
11.�
MIP
is a
30°
-60°
-90°
tria
ngl
e w
ith
rig
ht
angl
e at
I,a
nd
I�P�th
e lo
nge
r le
g.F
ind
the
coor
din
ates
of
Min
Qu
adra
nt
I fo
r I(
3,3)
an
d P
(12,
3).
( 3,3
�3�
3�)o
r ab
ou
t (3
,8.1
9)
12.�
TJ
Kis
a 4
5°-4
5°-9
0°tr
ian
gle
wit
h r
igh
t an
gle
at J
.Fin
d th
e co
ordi
nat
es o
f T
inQ
uad
ran
t II
for
J(�
2,�
3) a
nd
K(3
,�3)
.
(�2,
2)
13.B
OTA
NIC
AL
GA
RD
ENS
On
e of
th
e di
spla
ys a
t a
bota
nic
al g
arde
n
is a
n h
erb
gard
en p
lan
ted
in t
he
shap
e of
a s
quar
e.T
he
squ
are
mea
sure
s 6
yard
s on
eac
h s
ide.
Vis
itor
s ca
n v
iew
th
e h
erbs
fro
m a
diag
onal
pat
hw
ay t
hro
ugh
th
e ga
rden
.How
lon
g is
th
e pa
thw
ay?
6�2�
yd o
r ab
ou
t 8.
48 y
d
6 yd
6 yd
6 yd
6 yd
bA
B C
D
a
x
y60
�
30�
c
11�
2��
2
x
45�
11
y60
�3.
5
xy
x�y
28
9�2�
�2
9�2�
�2
y
x
30�
26y
x
2560�
yx
45�
9Pra
ctic
e (
Ave
rag
e)
Sp
ecia
l Rig
ht T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csS
pec
ial T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill36
7G
lenc
oe G
eom
etry
Lesson 7-3
Pre-
Act
ivit
yH
ow i
s tr
ian
gle
tili
ng
use
d i
n w
allp
aper
des
ign
?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-3
at
the
top
of p
age
357
in y
our
text
book
.•
How
can
you
mos
t co
mpl
etel
y de
scri
be t
he
larg
er t
rian
gle
and
the
two
smal
ler
tria
ngl
es i
n t
ile
15?
Sam
ple
an
swer
:Th
e la
rger
tri
ang
le is
an is
osc
eles
ob
tuse
tri
ang
le.T
he
two
sm
alle
r tr
ian
gle
s ar
eco
ng
ruen
t sc
alen
e ri
gh
t tr
ian
gle
s.•
How
can
you
mos
t co
mpl
etel
y de
scri
be t
he
larg
er t
rian
gle
and
the
two
smal
ler
tria
ngl
es i
n t
ile
16?
(In
clu
de a
ngl
e m
easu
res
in d
escr
ibin
g al
l th
etr
iang
les.
)S
amp
le a
nsw
er:T
he
larg
er t
rian
gle
is e
qu
ilate
ral,
soea
ch o
f it
s an
gle
mea
sure
s is
60.
Th
e tw
o s
mal
ler
tria
ng
les
are
con
gru
ent
rig
ht
tria
ng
les
in w
hic
h t
he
ang
le m
easu
res
are
30,6
0,an
d 9
0.
Rea
din
g t
he
Less
on
1.S
upp
ly t
he
corr
ect
nu
mbe
r or
nu
mbe
rs t
o co
mpl
ete
each
sta
tem
ent.
a.In
a 4
5°-4
5°-9
0°tr
ian
gle,
to f
ind
the
len
gth
of
the
hyp
oten
use
,mu
ltip
ly t
he
len
gth
of
a
leg
by
.
b.
In a
30°
-60°
-90°
tria
ngl
e,to
fin
d th
e le
ngt
h o
f th
e h
ypot
enu
se,m
ult
iply
th
e le
ngt
h o
f
the
shor
ter
leg
by
.
c.In
a 3
0°-6
0°-9
0°tr
iang
le,t
he lo
nger
leg
is o
ppos
ite
the
angl
e w
ith
a m
easu
re o
f .
d.
In a
30°
-60°
-90°
tria
ngl
e,to
fin
d th
e le
ngt
h o
f th
e lo
nge
r le
g,m
ult
iply
th
e le
ngt
h o
f
the
shor
ter
leg
by
.
e.In
an
iso
scel
es r
igh
t tr
ian
gle,
each
leg
is
oppo
site
an
an
gle
wit
h a
mea
sure
of
.
f.In
a 3
0°-6
0°-9
0°tr
ian
gle,
to f
ind
the
len
gth
of
the
shor
ter
leg,
divi
de t
he
len
gth
of
the
lon
ger
leg
by
.
g.In
30
°-60
°-90
°tr
ian
gle,
to f
ind
the
len
gth
of
the
lon
ger
leg,
divi
de t
he
len
gth
of
the
hyp
oten
use
by
and
mu
ltip
ly t
he
resu
lt b
y .
h.
To
fin
d th
e le
ngt
h o
f a
side
of
a sq
uar
e,di
vide
th
e le
ngt
h o
f th
e di
agon
al b
y .
2.In
dica
te w
het
her
eac
h s
tate
men
t is
alw
ays,
som
etim
es,o
r n
ever
tru
e.a.
Th
e le
ngt
hs
of t
he
thre
e si
des
of a
n i
sosc
eles
tri
angl
e sa
tisf
y th
e P
yth
agor
ean
Th
eore
m.
som
etim
esb
.T
he
len
gth
s of
th
e si
des
of a
30°
-60°
-90°
tria
ngl
e fo
rm a
Pyt
hag
orea
n t
ripl
e.n
ever
c.T
he
len
gth
s of
all
th
ree
side
s of
a 3
0°-6
0°-9
0°tr
ian
gle
are
posi
tive
in
tege
rs.
nev
er
Hel
pin
g Y
ou
Rem
emb
er3.
Som
e st
ude
nts
fin
d it
eas
ier
to r
emem
ber
mat
hem
atic
al c
once
pts
in t
erm
s of
spe
cifi
cn
um
bers
rat
her
th
an v
aria
bles
.How
can
you
use
spe
cifi
c n
um
bers
to
hel
p yo
u r
emem
ber
the
rela
tion
ship
bet
wee
n t
he
len
gth
s of
th
e th
ree
side
s in
a 3
0°-6
0°-9
0°tr
ian
gle?
Sam
ple
an
swer
:D
raw
a 3
0�-6
0�-9
0�tr
ian
gle
.Lab
el t
he
len
gth
of
the
sho
rter
leg
as
1.T
hen
th
e le
ng
th o
f th
e hy
po
ten
use
is 2
,an
d t
he
len
gth
of
the
lon
ger
leg
is �
3�.Ju
st r
emem
ber
:1,
2,�
3�.
�2�
�3�
2
�3�
45�
3�
602
�2�
©G
lenc
oe/M
cGra
w-H
ill36
8G
lenc
oe G
eom
etry
Co
nst
ruct
ing
Val
ues
of
Sq
uar
e R
oo
tsT
he
diag
ram
at
the
righ
t sh
ows
a ri
ght
isos
cele
s tr
ian
gle
wit
h
two
legs
of
len
gth
1 i
nch
.By
the
Pyt
hag
orea
n T
heo
rem
,th
e le
ngt
h
of t
he
hyp
oten
use
is
�2�
inch
es.B
y co
nst
ruct
ing
an a
djac
ent
righ
t tr
ian
gle
wit
h l
egs
of �
2�in
ches
an
d 1
inch
,you
can
cre
ate
a se
gmen
t of
len
gth
�3�.
By
con
tin
uin
g th
is p
roce
ss a
s sh
own
bel
ow,y
ou c
an c
onst
ruct
a
“wh
eel”
of s
quar
e ro
ots.
Th
is w
hee
l is
cal
led
the
“Wh
eel
of T
heo
doru
s”af
ter
a G
reek
ph
ilos
oph
er w
ho
live
d ab
out
400
B.C
.
Con
tin
ue
con
stru
ctin
g th
e w
hee
l u
nti
l yo
u m
ake
a se
gmen
t of
len
gth
�18�
.
��
1
1
1
3�
�
2
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-3
7-3
1
1
11
1
1��2
��3
��5
��6
��7 ��8
��10
��11
��12
��13
��14
��15
��17��18
��16
� 4
��4
� 2
��9
� 3
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Trig
on
om
etry
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill36
9G
lenc
oe G
eom
etry
Lesson 7-4
Trig
on
om
etri
c R
atio
sT
he
rati
o of
th
e le
ngt
hs
of t
wo
side
s of
a r
igh
t tr
ian
gle
is c
alle
d a
trig
onom
etri
c ra
tio.
Th
e th
ree
mos
t co
mm
on r
atio
s ar
e si
ne,
cosi
ne,
and
tan
gen
t,w
hic
h a
re a
bbre
viat
ed s
in,c
os,a
nd
tan
,re
spec
tive
ly.
sin
R��le
g hyop pp oto es nit ue s� e
R�
cos
R�
tan
R�
��r t�
��s t�
��r s�
Fin
d s
in A
,cos
A,a
nd
tan
A.E
xpre
ss e
ach
rat
io a
s
a d
ecim
al t
o th
e n
eare
st t
hou
san
dth
.
sin
A��o hp yp po os ti et ne ul se eg
�co
s A
��a hd yj pa oc te en nt ul se eg
�ta
n A
�� aop dp jao cs ei nte t
l le eg g�
��B A
BC ��
� AABC �
��B A
CC �
�� 15 3�
��1 12 3�
�� 15 2�
�0.
385
�0.
923
�0.
417
Fin
d t
he
ind
icat
ed t
rigo
nom
etri
c ra
tio
as a
fra
ctio
n
and
as
a d
ecim
al.I
f n
eces
sary
,rou
nd
to
the
nea
rest
te
n-t
hou
san
dth
.
1.si
n A
2.ta
n B
�1 15 7�;
0.88
24� 18 5�
;0.
5333
3.co
s A
4.co
s B
� 18 7�;
0.47
06�1 15 7�
;0.
8824
5.si
n D
6.ta
n E
�4 5� ;0.
8�3 4� ;
0.75
7.co
s E
8.co
s D
�4 5� ;0.
8�3 5� ;
0.6
16
1620
12
3430 CB
AD
FE
12135 CB
A
leg
oppo
site
�R
��
�le
g ad
jace
nt
to �
Rle
g ad
jace
nt
to �
R�
��
hyp
oten
use
str TS
R
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill37
0G
lenc
oe G
eom
etry
Use
Tri
go
no
met
ric
Rat
ios
In a
rig
ht
tria
ngl
e,if
you
kn
ow t
he
mea
sure
s of
tw
o si
des
or i
f yo
u k
now
th
e m
easu
res
of o
ne
side
an
d an
acu
te a
ngl
e,th
en y
ou c
an u
se t
rigo
nom
etri
cra
tios
to
fin
d th
e m
easu
res
of t
he
mis
sin
g si
des
or a
ngl
es o
f th
e tr
ian
gle.
Fin
d x
,y,a
nd
z.R
oun
d e
ach
mea
sure
to
the
nea
rest
w
hol
e n
um
ber
.18
58�
x�C
Byz
A
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Trig
on
om
etry
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
a.F
ind
x.
x�
58�
90x
�32
b.
Fin
d y
.
tan
A�
� 1y 8�
tan
58°
�� 1y 8�
y�
18 t
an 5
8°y
�29
c.F
ind
z.
cos
A�
�1 z8 �
cos
58°
��1 z8 �
zco
s 58
°�
18
z�
� cos18
58°
�
z�
34
Exer
cises
Exer
cises
Fin
d x
.Rou
nd
to
the
nea
rest
ten
th.
1.2.
17.0
48.6
3.4.
22.6
76.0
5.6.
24.9
34.2
1564
�x
16
40�
x
4
1x�
12
5x�
1216 x�
3228
�x
Exam
ple
Exam
ple
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Trig
on
om
etry
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill37
1G
lenc
oe G
eom
etry
Lesson 7-4
Use
�R
ST
to f
ind
sin
R,c
os R
,tan
R,s
in S
,cos
S,a
nd
tan
S.
Exp
ress
eac
h r
atio
as
a fr
acti
on a
nd
as
a d
ecim
al t
o th
e n
eare
st h
un
dre
dth
.
1.r
�16
,s�
30,t
�34
2.r
�10
,s�
24,t
�26
sin
R�
�1 36 4��
0.47
;si
n R
��1 20 6�
�0.
38;
cos
R�
�3 30 4��
0.88
;co
s R
��2 24 6�
�0.
92;
tan
R�
�1 36 0��
0.53
;ta
n R
��1 20 4�
�0.
42;
sin
S�
�3 30 4��
0.88
;si
n S
��2 24 6�
�0.
92;
cos
S�
�1 36 4��
0.47
;co
s S
��1 20 6�
�0.
38;
tan
S�
�3 10 6��
1.88
tan
S�
�2 14 0��
2.4
Use
a c
alcu
lato
r to
fin
d e
ach
val
ue.
Rou
nd
to
the
nea
rest
ten
-th
ousa
nd
th.
3.si
n 5
0.08
724.
tan
23
0.42
455.
cos
610.
4848
6.si
n 7
5.8
0.96
947.
tan
17.
30.
3115
8.co
s 52
.90.
6032
Use
th
e fi
gure
to
fin
d e
ach
tri
gon
omet
ric
rati
o.E
xpre
ss
answ
ers
as a
fra
ctio
n a
nd
as
a d
ecim
al r
oun
ded
to
the
nea
rest
ten
-th
ousa
nd
th.
9.ta
n C
10.s
in A
11.c
os C
� 49 0��
0.22
50�4 40 1�
�0.
9756
�4 40 1��
0.97
56
Fin
d t
he
mea
sure
of
each
acu
te a
ngl
e to
th
e n
eare
st t
enth
of
a d
egre
e.
12.s
in B
�0.
2985
17.4
13.t
an A
�0.
4168
22.6
14.c
os R
�0.
8443
32.4
15.t
an C
�0.
3894
21.3
16.c
os B
�0.
7329
42.9
17.s
in A
�0.
1176
6.8
Fin
d x
.Rou
nd
to
the
nea
rest
ten
th.
18.
19.
20.
28.8
73.5
15.9
19
x
33�
UL
S
27
x�8
BAC
27
x�
13 BA
C
41
409
B
A
C
sR
S Trt
©G
lenc
oe/M
cGra
w-H
ill37
2G
lenc
oe G
eom
etry
Use
�L
MN
to f
ind
sin
L,c
os L
,tan
L,s
in M
,cos
M,a
nd
tan
M.
Exp
ress
eac
h r
atio
as
a fr
acti
on a
nd
as
a d
ecim
al t
o th
e n
eare
st h
un
dre
dth
.
1.�
�15
,m�
36,n
�39
2.�
�12
,m�
12�
3�,n
�24
sin
L�
�1 35 9��
0.38
;si
n L
��1 22 4�
�0.
50;
cos
L�
�3 36 9��
0.92
;co
s L
��
0.87
;
tan
L�
�1 35 6��
0.42
;ta
n L
��
0.58
;
sin
M�
�3 36 9��
0.92
;si
n M
��
0.87
;
cos
M�
�1 35 9��
0.38
;co
s M
��1 22 4�
�0.
50;
tan
M�
�3 16 5��
2.4
tan
M�
�1.
73
Use
a c
alcu
lato
r to
fin
d e
ach
val
ue.
Rou
nd
to
the
nea
rest
ten
-th
ousa
nd
th.
3.si
n 9
2.4
0.99
914.
tan
27.
50.
5206
5.co
s 64
.80.
4258
Use
th
e fi
gure
to
fin
d e
ach
tri
gon
omet
ric
rati
o.E
xpre
ss
answ
ers
as a
fra
ctio
n a
nd
as
a d
ecim
al r
oun
ded
to
the
nea
rest
ten
-th
ousa
nd
th.
6.co
s A
7.ta
n B
8.si
n A
�0.
9487
�3 1��
3.00
00�
0.31
62
Fin
d t
he
mea
sure
of
each
acu
te a
ngl
e to
th
e n
eare
st t
enth
of
a d
egre
e.
9.si
n B
�0.
7823
51.5
10.t
an A
�0.
2356
13.3
11.c
os R
�0.
6401
50.2
Fin
d x
.Rou
nd
to
the
nea
rest
ten
th.
12.
64.4
13.
18.1
14.
24.2
15.G
EOG
RA
PHY
Die
go u
sed
a th
eodo
lite
to
map
a r
egio
n o
f la
nd
for
his
cl
ass
in g
eom
orph
olog
y.T
o de
term
ine
the
elev
atio
n o
f a
vert
ical
roc
kfo
rmat
ion
,he
mea
sure
d th
e di
stan
ce f
rom
th
e ba
se o
f th
e fo
rmat
ion
to
his
pos
itio
n a
nd
the
angl
e be
twee
n t
he
grou
nd
and
the
lin
e of
sig
ht
to
the
top
of t
he
form
atio
n.T
he
dist
ance
was
43
met
ers
and
the
angl
e w
as
36 d
egre
es.W
hat
is
the
hei
ght
of t
he
form
atio
n t
o th
e n
eare
st m
eter
?31
m
36� 43
m
41�
x
3229
x�9
23
x�
11
�10�
�10
3 �10�
�10
15
5 ��10
5 CA
B
12�
3��
12
12�
3��
2412� 12
�3�
12�
3��
24
ML
N
Pra
ctic
e (
Ave
rag
e)
Trig
on
om
etry
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csTr
igo
no
met
ry
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill37
3G
lenc
oe G
eom
etry
Lesson 7-4
Pre-
Act
ivit
yH
ow c
an s
urv
eyor
s d
eter
min
e an
gle
mea
sure
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-4
at
the
top
of p
age
364
in y
our
text
book
.
•W
hy
is i
t im
port
ant
to d
eter
min
e th
e re
lati
ve p
osit
ion
s ac
cura
tely
in
nav
igat
ion
? (G
ive
two
poss
ible
rea
son
s.)
Sam
ple
an
swer
s:(1
) To
avo
id c
olli
sio
ns
bet
wee
n s
hip
s,an
d (
2) t
o p
reve
nt
ship
sfr
om
losi
ng
th
eir
bea
rin
gs
and
get
tin
g lo
st a
t se
a.•
Wh
at d
oes
cali
brat
edm
ean
? S
amp
le a
nsw
er:
mar
ked
pre
cise
ly t
op
erm
it a
ccu
rate
mea
sure
men
ts t
o b
e m
ade
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.W
rite
a r
atio
usi
ng
the
side
len
gth
s in
th
e
figu
re t
o re
pres
ent
each
of
the
foll
owin
g tr
igon
omet
ric
rati
os.
A.
sin
N� MM
NP �B
.cos
N� MN
P N�
C.
tan
N�M N
PP �D
.tan
M� MN
P P�
E.
sin
M� MN
P N�F.
cos
M� MM
NP �
2.A
ssu
me
that
you
en
ter
each
of
the
expr
essi
ons
in t
he
list
on
th
e le
ft i
nto
you
r ca
lcu
lato
r.M
atch
eac
h o
f th
ese
expr
essi
ons
wit
h a
des
crip
tion
fro
m t
he
list
on
th
e ri
ght
to t
ell
wh
atyo
u a
re f
indi
ng
wh
en y
ou e
nte
r th
is e
xpre
ssio
n.
P
MN
a.si
n 2
0v
b.
cos
20ii
c.si
n�
10.
8vi
d.
tan
�1
0.8
iiie.
tan
20
ivf.
cos�
10.
8i
i.th
e de
gree
mea
sure
of
an a
cute
an
gle
wh
ose
cosi
ne
is 0
.8
ii.
the
rati
o of
th
e le
ngt
h o
f th
e le
g ad
jace
nt
to t
he
20°
angl
e to
th
ele
ngt
h o
f h
ypot
enu
se i
n a
20°
-70°
-90°
tria
ngl
e
iii.
the
degr
ee m
easu
re o
f an
acu
te a
ngl
e in
a r
igh
t tr
ian
gle
for
wh
ich
th
e ra
tio
of t
he
len
gth
of
the
oppo
site
leg
to
the
len
gth
of
the
adja
cen
t le
g is
0.8
iv.t
he r
atio
of
the
leng
th o
f th
e le
g op
posi
te t
he 2
0°an
gle
to t
hele
ngth
of
the
leg
adja
cent
to
it i
n a
20°-
70°-
90°
tria
ngle
v.th
e ra
tio
of t
he
len
gth
of
the
leg
oppo
site
th
e 20
°an
gle
to t
he
len
gth
of
hyp
oten
use
in
a 2
0°-7
0°-9
0°tr
ian
gle
vi.t
he
degr
ee m
easu
re o
f an
acu
te a
ngl
e in
a r
igh
t tr
ian
gle
for
wh
ich
th
e ra
tio
of t
he
len
gth
of
the
oppo
site
leg
to
the
len
gth
of
the
hyp
oten
use
is
0.8
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an t
he
coin
cos
ine
hel
p yo
u t
o re
mem
ber
the
rela
tion
ship
bet
wee
n t
he
sin
es a
nd
cosi
nes
of
the
two
acu
te a
ngl
es o
f a
righ
t tr
ian
gle?
Sam
ple
an
swer
:Th
e co
in c
osi
ne
com
es f
rom
co
mp
lem
ent,
as in
com
ple
men
tary
ang
les.
Th
e co
sin
e o
f an
acu
te a
ng
le is
eq
ual
to
th
e si
ne
of
its
com
ple
men
t.
©G
lenc
oe/M
cGra
w-H
ill37
4G
lenc
oe G
eom
etry
Sin
e an
d C
osi
ne
of
An
gle
sT
he
foll
owin
g di
agra
m c
an b
e u
sed
to o
btai
n a
ppro
xim
ate
valu
es f
or t
he
sin
ean
d co
sin
e of
an
gles
fro
m 0
°to
90°
.Th
e ra
diu
s of
th
e ci
rcle
is
1.S
o,th
e si
ne
and
cosi
ne
valu
es c
an b
e re
ad d
irec
tly
from
th
e ve
rtic
al a
nd
hor
izon
tal
axes
.
Fin
d a
pp
roxi
mat
e va
lues
for
sin
40°
and
cos
40�
.Con
sid
er t
he
tria
ngl
e fo
rmed
by
the
segm
ent
mar
ked
40°
,as
illu
stra
ted
by
the
shad
ed
tria
ngl
e at
rig
ht.
sin
40°
��a c�
��0.
164 �or
0.6
4co
s 40
°�
�b c��
�0.177 �
or 0
.77
1.U
se t
he
diag
ram
abo
ve t
o co
mpl
ete
the
char
t of
val
ues
.
2.C
ompa
re t
he
sin
e an
d co
sin
e of
tw
o co
mpl
emen
tary
an
gles
(an
gles
wh
ose
sum
is
90°)
.Wh
at d
o yo
u n
otic
e?
Th
e si
ne
of
an a
ng
le is
eq
ual
to
th
e co
sin
e o
f th
e co
mp
lem
ent
of
the
ang
le.
00.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.91
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
90°
0°
10°
20°
30°
40°
50°
60°
70°
80°
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-4
7-4 x
°0°
10°
20°
30°
40°
50°
60°
70°
80°
90°
sin
x°
00.
170.
340.
50.
640.
770.
870.
940.
981
cos
x°
10.
980.
940.
870.
770.
640.
50.
340.
170
1 0
40°
0.64
c �
1 u
nit
x°b
� c
os x
°0.
771
a �
sin
x°
Exam
ple
Exam
ple
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
An
gle
s o
f E
leva
tio
n a
nd
Dep
ress
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill37
5G
lenc
oe G
eom
etry
Lesson 7-5
An
gle
s o
f El
evat
ion
Man
y re
al-w
orld
pro
blem
s th
at i
nvo
lve
look
ing
up
to a
n o
bjec
t ca
n b
e de
scri
bed
in t
erm
s of
an
an
gle
of
elev
atio
n,w
hic
h i
s th
e an
gle
betw
een
an
obs
erve
r’s
lin
e of
sig
ht
and
a h
oriz
onta
l li
ne.
Th
e an
gle
of e
leva
tion
fro
m p
oin
t A
to t
he
top
of
a c
liff
is
34°.
If p
oin
t A
is 1
000
feet
fro
m t
he
bas
e of
th
e cl
iff,
how
hig
h i
s th
e cl
iff?
Let
x�
the
hei
ght
of t
he
clif
f.
tan
34°
�� 10
x 00�ta
n �
�o ap dp jao cs eit ne t�
1000
(tan
34°
)�
xM
ultip
ly e
ach
side
by
1000
.
674.
5�
xU
se a
cal
cula
tor.
Th
e h
eigh
t of
th
e cl
iff
is a
bou
t 67
4.5
feet
.
Sol
ve e
ach
pro
ble
m.R
oun
d m
easu
res
of s
egm
ents
to
the
nea
rest
wh
ole
nu
mb
eran
d a
ngl
es t
o th
e n
eare
st d
egre
e.
1.T
he
angl
e of
ele
vati
on f
rom
poi
nt
Ato
th
e to
p of
a h
ill
is 4
9°.
If p
oin
t A
is 4
00 f
eet
from
th
e ba
se o
f th
e h
ill,
how
hig
h i
s th
e h
ill?
460
ft
2.F
ind
the
angl
e of
ele
vati
on o
f th
e su
n w
hen
a 1
2.5-
met
er-t
all
tele
phon
e po
le c
asts
a 1
8-m
eter
-lon
g sh
adow
.
35°
3.A
lad
der
lean
ing
agai
nst
a b
uil
din
g m
akes
an
an
gle
of 7
8°w
ith
th
e gr
oun
d.T
he
foot
of
the
ladd
er i
s 5
feet
fro
m t
he
buil
din
g.H
ow l
ong
is t
he
ladd
er?
24 f
t
4.A
per
son
wh
ose
eyes
are
5 f
eet
abov
e th
e gr
oun
d is
sta
ndi
ng
on t
he
run
way
of
an a
irpo
rt 1
00 f
eet
from
th
e co
ntr
ol t
ower
.T
hat
per
son
obs
erve
s an
air
tra
ffic
con
trol
ler
at t
he
win
dow
of
th
e 13
2-fo
ot t
ower
.Wh
at i
s th
e an
gle
of e
leva
tion
?
52°
?5
ft10
0 ft
132
ft
78�
5 ft
?
18 m
12.5
msun
?
✹
400
ft
?
49�
A
?
1000
ft34
�A
x
angl
e of
elev
atio
n
line o
f sigh
t
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill37
6G
lenc
oe G
eom
etry
An
gle
s o
f D
epre
ssio
nW
hen
an
obs
erve
r is
loo
kin
g do
wn
,th
e an
gle
of d
epre
ssio
nis
th
e an
gle
betw
een
th
e ob
serv
er’s
lin
e of
sig
ht
and
a h
oriz
onta
l li
ne.
Th
e an
gle
of d
epre
ssio
n f
rom
th
e to
p o
f an
80
-foo
t b
uil
din
g to
poi
nt
Aon
th
e gr
oun
d i
s 42
°.H
ow f
ar
is t
he
foot
of
the
bu
ild
ing
from
poi
nt
A?
Let
x�
the
dist
ance
fro
m p
oin
t A
to t
he
foot
of
the
buil
din
g.S
ince
th
e h
oriz
onta
l li
ne
is p
aral
lel
to t
he
grou
nd,
the
angl
e of
dep
ress
ion
�D
BA
is c
ongr
uen
t to
�B
AC
.
tan
42°
��8 x0 �
tan
��o ap dp jao cs eit ne t
�
x(ta
n 4
2°)
�80
Mul
tiply
eac
h si
de b
y x.
x�
� tan80
42°
�D
ivid
e ea
ch s
ide
by t
an 4
2°.
x�
88.8
Use
a c
alcu
lato
r.
Poi
nt
Ais
abo
ut
89 f
eet
from
th
e ba
se o
f th
e bu
ildi
ng.
Sol
ve e
ach
pro
ble
m.R
oun
d m
easu
res
of s
egm
ents
to
the
nea
rest
wh
ole
nu
mb
eran
d a
ngl
es t
o th
e n
eare
st d
egre
e.
1.T
he
angl
e of
dep
ress
ion
fro
m t
he
top
of a
sh
eer
clif
f to
po
int
Aon
th
e gr
oun
d is
35°
.If
poin
t A
is 2
80 f
eet
from
th
e ba
se o
f th
e cl
iff,
how
tal
l is
th
e cl
iff?
196
ft
2.T
he
angl
e of
dep
ress
ion
fro
m a
bal
loon
on
a 7
5-fo
ot
stri
ng
to a
per
son
on
th
e gr
oun
d is
36°
.How
hig
h i
s th
e ba
lloo
n?
44 f
t
3.A
ski
ru
n i
s 10
00 y
ards
lon
g w
ith
a v
erti
cal
drop
of
208
yard
s.F
ind
the
angl
e of
dep
ress
ion
fro
m t
he
top
of t
he
ski
run
to
the
bott
om.
12°
4.F
rom
th
e to
p of
a 1
20-f
oot-
hig
h t
ower
,an
air
tra
ffic
co
ntr
olle
r ob
serv
es a
n a
irpl
ane
on t
he
run
way
at
an
angl
e of
dep
ress
ion
of
19°.
How
far
fro
m t
he
base
of
the
tow
er i
s th
e ai
rpla
ne?
349
ft
120
ft
?
19�
208
yd
?
1000
yd
36�
75 ft
?
A
35�
280
ft
?
ACB
D
x42
�
angl
e of
depr
essi
on
horiz
onta
l
80 ft
Ylin
e of s
ight
horiz
onta
lan
gle
ofde
pres
sion
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
An
gle
s o
f E
leva
tio
n a
nd
Dep
ress
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
An
gle
s o
f E
leva
tio
n a
nd
Dep
ress
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill37
7G
lenc
oe G
eom
etry
Lesson 7-5
Nam
e th
e an
gle
of d
epre
ssio
n o
r an
gle
of e
leva
tion
in
eac
h f
igu
re.
1.2.
�F
LS
;�
TS
L�
RT
W;
�S
WT
3.4.
�D
CB
;�
AB
C�
WZ
P;
�R
PZ
5.M
OU
NTA
IN B
IKIN
GO
n a
mou
nta
in b
ike
trip
alo
ng
the
Gem
ini
Bri
dges
Tra
il i
n M
oab,
Uta
h,N
abu
ko s
topp
ed o
n t
he
can
yon
flo
or t
o ge
t a
good
vie
w o
f th
e tw
in s
ands
ton
ebr
idge
s.N
abu
ko i
s st
andi
ng
abou
t 60
met
ers
from
th
e ba
se o
f th
e ca
nyo
n c
liff
,an
d th
en
atu
ral
arch
bri
dges
are
abo
ut
100
met
ers
up
the
can
yon
wal
l.If
her
lin
e of
sig
ht
is f
ive
feet
abo
ve t
he
grou
nd,
wh
at i
s th
e an
gle
of e
leva
tion
to
the
top
of t
he
brid
ges?
Rou
nd
toth
e n
eare
st t
enth
deg
ree.
abo
ut
57.7
�
6.SH
AD
OW
SS
upp
ose
the
sun
cas
ts a
sh
adow
off
a 3
5-fo
ot b
uil
din
g.If
th
e an
gle
of e
leva
tion
to
the
sun
is
60°,
how
lon
g is
th
e sh
adow
to
th
e n
eare
st t
enth
of
a fo
ot?
abo
ut
20.2
ft
7.B
ALL
OO
NIN
GF
rom
her
pos
itio
n i
n a
hot
-air
bal
loon
,An
gie
can
see
her
car
par
ked
in a
fiel
d.If
th
e an
gle
of d
epre
ssio
n i
s 8°
and
An
gie
is 3
8 m
eter
s ab
ove
the
grou
nd,
wh
at i
sth
e st
raig
ht-
lin
e di
stan
ce f
rom
An
gie
to h
er c
ar?
Rou
nd
to t
he
nea
rest
wh
ole
met
er.
abo
ut
273
m
8.IN
DIR
ECT
MEA
SUR
EMEN
TK
yle
is a
t th
e en
d of
a p
ier
30 f
eet
abov
e th
e oc
ean
.His
eye
lev
el i
s 3
feet
abo
ve t
he
pier
.He
is u
sin
g bi
noc
ula
rs t
o w
atch
a w
hal
e su
rfac
e.If
th
e an
gle
of d
epre
ssio
n
of t
he
wh
ale
is 2
0°,h
ow f
ar i
s th
e w
hal
e fr
om
Kyl
e’s
bin
ocu
lars
? R
oun
d to
th
e n
eare
st t
enth
foo
t.
abo
ut
96.5
ft
wha
lew
ater
leve
l
20�
Kyle
’s ey
es
pier
3 ft
30 ft
60� ?
35 ft
Z
PW
R
D
AC
B
T
WR
S
F
T
L
S
©G
lenc
oe/M
cGra
w-H
ill37
8G
lenc
oe G
eom
etry
Nam
e th
e an
gle
of d
epre
ssio
n o
r an
gle
of e
leva
tion
in
eac
h f
igu
re.
1.2.
�T
RZ
;�
YZ
R�
PR
M;
�L
MR
3.W
ATE
R T
OW
ERS
A s
tude
nt
can
see
a w
ater
tow
er f
rom
th
e cl
oses
t po
int
of t
he
socc
erfi
eld
at S
an L
obos
Hig
h S
choo
l.T
he
edge
of
the
socc
er f
ield
is
abou
t 11
0 fe
et f
rom
th
ew
ater
tow
er a
nd
the
wat
er t
ower
sta
nds
at
a h
eigh
t of
32.
5 fe
et.W
hat
is
the
angl
e of
elev
atio
n i
f th
e ey
e le
vel
of t
he
stu
den
t vi
ewin
g th
e to
wer
fro
m t
he
edge
of
the
socc
erfi
eld
is 6
fee
t ab
ove
the
grou
nd?
Rou
nd
to t
he
nea
rest
ten
th d
egre
e.
abo
ut
13.5
�
4.C
ON
STR
UC
TIO
NA
roo
fer
prop
s a
ladd
er a
gain
st a
wal
l so
th
at t
he
top
of t
he
ladd
erre
ach
es a
30-
foot
roo
f th
at n
eeds
rep
air.
If t
he
angl
e of
ele
vati
on f
rom
th
e bo
ttom
of
the
ladd
er t
o th
e ro
of i
s 55
°,h
ow f
ar i
s th
e la
dder
fro
m t
he
base
of
the
wal
l? R
oun
d yo
ur
answ
er t
o th
e n
eare
st f
oot.
abo
ut
21 f
t
5.TO
WN
OR
DIN
AN
CES
Th
e to
wn
of
Bel
mon
t re
stri
cts
the
hei
ght
of f
lagp
oles
to
25 f
eet
on a
ny
prop
erty
.Lin
dsay
wan
ts t
o de
term
ine
wh
eth
er h
er s
choo
l is
in
com
plia
nce
wit
h t
he
regu
lati
on.H
er e
ye
leve
l is
5.5
fee
t fr
om t
he
grou
nd
and
she
stan
ds 3
6 fe
et f
rom
th
efl
agpo
le.I
f th
e an
gle
of e
leva
tion
is
abou
t 25
°,w
hat
is
the
hei
ght
of t
he
flag
pole
to
the
nea
rest
ten
th f
oot?
abo
ut
22.3
ft
6.G
EOG
RA
PHY
Ste
phan
is
stan
din
g on
a m
esa
at t
he
Pai
nte
d D
eser
t.T
he
elev
atio
n o
fth
e m
esa
is a
bou
t 13
80 m
eter
s an
d S
teph
an’s
eye
lev
el i
s 1.
8 m
eter
s ab
ove
grou
nd.
IfS
teph
an c
an s
ee a
ban
d of
mu
ltic
olor
ed s
hal
e at
th
e bo
ttom
an
d th
e an
gle
of d
epre
ssio
nis
29°
,abo
ut
how
far
is
the
ban
d of
sh
ale
from
his
eye
s? R
oun
d to
th
e n
eare
st m
eter
.
abo
ut
2850
m
7.IN
DIR
ECT
MEA
SUR
EMEN
TM
r.D
omin
guez
is
stan
din
g on
a 4
0-fo
ot o
cean
blu
ff n
ear
his
hom
e.H
e ca
n s
ee h
is t
wo
dogs
on
th
e be
ach
bel
ow.I
f h
is l
ine
of s
igh
t is
6 f
eet
abov
e th
e gr
oun
d an
d th
e an
gles
of
depr
essi
on t
o h
is d
ogs
are
34°
and
48°,
how
far
apa
rt a
re t
he
dogs
to
the
nea
rest
foo
t?
abo
ut
27 f
t48
�34
�
40 ft
6 ft
Mr.
Dom
ingu
ez
bluf
f
25�
5.5
ft36
ft
x
R
M
P
L
T
YR
Z
Pra
ctic
e (
Ave
rag
e)
An
gle
s o
f E
leva
tio
n a
nd
Dep
ress
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csA
ng
les
of
Ele
vati
on
an
d D
epre
ssio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill37
9G
lenc
oe G
eom
etry
Lesson 7-5
Pre-
Act
ivit
yH
ow d
o ai
rlin
e p
ilot
s u
se a
ngl
es o
f el
evat
ion
an
d d
epre
ssio
n?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-5
at
the
top
of p
age
371
in y
our
text
book
.
Wh
at d
oes
the
angl
e m
easu
re t
ell
the
pilo
t?S
amp
le a
nsw
er:
ho
wst
eep
her
asc
ent
mu
st b
e to
cle
ar t
he
pea
k
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.T
he
two
obse
rver
s ar
e lo
okin
g at
on
e an
othe
r.S
elec
t th
e co
rrec
t ch
oice
for
eac
h qu
esti
on.
a.W
hat
is
the
lin
e of
sig
ht?
iii(i
) li
ne
RS
(ii)
lin
e S
T(i
ii)
lin
e R
T(i
v) l
ine
TU
b.
Wh
at i
s th
e an
gle
of e
leva
tion
?ii
(i)
�R
ST
(ii)
�S
RT
(iii
) �
RT
S(i
v) �
UT
R
c.W
hat
is
the
angl
e of
dep
ress
ion
?iv
(i)
�R
ST
(ii)
�S
RT
(iii
) �
RT
S(i
v) �
UT
R
d.
How
are
th
e an
gle
of e
leva
tion
an
d th
e an
gle
of d
epre
ssio
n r
elat
ed?
ii(i
)T
hey
are
com
plem
enta
ry.
(ii)
Th
ey a
re c
ongr
uen
t.(i
ii)
Th
ey a
re s
upp
lem
enta
ry.
(iv)
Th
e an
gle
of e
leva
tion
is
larg
er t
han
th
e an
gle
of d
epre
ssio
n.
e.W
hic
h p
ostu
late
or
theo
rem
th
at y
ou l
earn
ed i
n C
hap
ter
3 su
ppor
ts y
our
answ
er f
orpa
rt c
?iv
(i)
Cor
resp
ondi
ng
An
gles
Pos
tula
te(i
i)A
lter
nat
e E
xter
ior
An
gles
Th
eore
m(i
ii)
Con
secu
tive
In
teri
or A
ngl
es T
heo
rem
(iv)
Alt
ern
ate
Inte
rior
An
gles
Th
eore
m
2.A
stu
den
t sa
ys t
hat
th
e an
gle
of e
leva
tion
fro
m h
is e
ye t
o th
e to
p of
a f
lagp
ole
is 1
35°.
Wh
at i
s w
ron
g w
ith
th
e st
ude
nt’s
sta
tem
ent?
An
an
gle
of
elev
atio
n c
ann
ot
be
ob
tuse
.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
som
eth
ing
is t
o ex
plai
n i
t to
som
eon
e el
se.S
upp
ose
a cl
assm
ate
find
s it
dif
ficu
lt t
o di
stin
guis
h be
twee
n an
gles
of
elev
atio
n an
d an
gles
of
depr
essi
on.W
hat
are
som
e h
ints
you
can
giv
e h
er t
o h
elp
her
get
it
righ
t ev
ery
tim
e?S
amp
le a
nsw
ers:
(1) T
he
ang
le o
f d
epre
ssio
n a
nd
th
e an
gle
of
elev
atio
n a
re b
oth
mea
sure
db
etw
een
th
e h
ori
zon
tal a
nd
th
e lin
e o
f si
gh
t.(2
) Th
e an
gle
of
dep
ress
ion
is a
lway
s co
ng
ruen
t to
th
e an
gle
of
elev
atio
n in
th
e sa
me
dia
gra
m.
(3)
Ass
oci
ate
the
wo
rd e
leva
tio
nw
ith
th
e w
ord
up
and
th
e w
ord
dep
ress
ion
wit
h t
he
wo
rd d
own
.
STob
serv
er a
tto
p of
bui
ldin
g
obse
rver
on g
roun
dR
U
©G
lenc
oe/M
cGra
w-H
ill38
0G
lenc
oe G
eom
etry
Rea
din
g M
ath
emat
ics
Th
e th
ree
mos
t co
mm
on t
rigo
nom
etri
c ra
tios
are
si
ne,
cosi
ne,
and
tan
gen
t.T
hre
e ot
her
rat
ios
are
the
cose
can
t,se
can
t,an
d co
tan
gen
t.T
he
char
t be
low
sh
ows
abbr
evia
tion
s an
d de
fin
itio
ns
for
all
six
rati
os.
Ref
er t
o th
e tr
ian
gle
at t
he
righ
t.
Use
th
e ab
bre
viat
ion
s to
rew
rite
eac
h s
tate
men
t as
an
eq
uat
ion
.
1.T
he
seca
nt
of a
ngl
e A
is
equ
al t
o 1
divi
ded
by t
he
cosi
ne
of a
ngl
e A
.se
c A
�� co
1 sA
�
2.T
he
cose
can
t of
an
gle
A i
s eq
ual
to
1 di
vide
d by
th
e si
ne
of a
ngl
e A
.cs
c A
�� si
n1A�
3.T
he
cota
nge
nt
of a
ngl
e A
is
equ
al t
o 1
divi
ded
by t
he
tan
gen
t of
an
gle
A.
cot
A�
� tan1
A�
4.T
he c
osec
ant
of a
ngle
A m
ulti
plie
d by
the
sin
e of
ang
le A
is e
qual
to
1.cs
c A
sin
A�
1
5.T
he s
ecan
t of
ang
le A
mul
tipl
ied
by t
he c
osin
e of
ang
le A
is e
qual
to
1.se
c A
co
s A
�1
6.T
he
cota
nge
nt
of a
ngl
e A
tim
es t
he
tan
gen
t of
an
gle
Ais
equ
al t
o 1.
cot
A t
an A
�1
Use
th
e tr
ian
gle
at r
igh
t.W
rite
eac
h r
atio
.
7.se
c R
� st �8.
csc
R� rt �
9.co
t R
�s r�
10.s
ec S
� rt �11
.cs
c S
� st �12
.co
t S
� sr �
13.I
f si
n x
°�
0.28
9,fi
nd
the
valu
e of
csc
x°.
�3.
46
14.I
f ta
n x
°�
1.37
6,fi
nd
the
valu
e of
cot
x°.
�0.
727
R TS
ts
r
A
ca
b
B C
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-5
7-5
Ab
bre
viat
ion
Rea
d a
s:R
atio
sin
Ath
e si
ne o
f �
A�
�a c�
cos
Ath
e co
sine
of
�A
��b c�
tan
Ath
e ta
ngen
t of
�A
��a b�
csc
Ath
e co
seca
nt o
f �
A�
� ac �
sec
Ath
e se
cant
of
�A
�� bc �
cot
Ath
e co
tang
ent
of �
A�
�b a�le
gad
jace
nt t
o�
A�
��
leg
oppo
site
�A
hypo
tenu
se�
��
leg
adja
cent
to
�A
hypo
tenu
se�
�le
gop
posi
te�
A
leg
oppo
site
�A
��
�le
gad
jace
nt t
o�
A
leg
adja
cent
to
�A
��
�hy
pote
nuse
leg
oppo
site
�A
��
hypo
tenu
se
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Th
e L
aw o
f S
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-H
ill38
1G
lenc
oe G
eom
etry
Lesson 7-6
The
Law
of
Sin
esIn
an
y tr
ian
gle,
ther
e is
a s
peci
al r
elat
ion
ship
bet
wee
n t
he
angl
es o
fth
e tr
ian
gle
and
the
len
gth
s of
th
e si
des
oppo
site
th
e an
gles
.
Law
of
Sin
es�si
n aA �
��si
n bB �
��si
n cC �
In �
AB
C,f
ind
b.
�sin c
C ��
�sin b
B �La
w o
f S
ines
�sin 30
45°
��
�sin b74
°�
m�
C�
45, c
�30
, m�
B�
74
bsi
n 4
5°�
30 s
in 7
4°C
ross
mul
tiply.
b�
�30si
s nin 457 °4°
�D
ivid
e ea
ch s
ide
by s
in 4
5°.
b�
40.8
Use
a c
alcu
lato
r.
45�
3074
�
bB
AC
In �
DE
F,f
ind
m�
D.
�sin d
D ��
�sin e
E �La
w o
f S
ines
�si2n 8D �
��si
n 2458
°�
d�
28
, m
�E
�5
8,
e�
24
24 s
in D
�28
sin
58°
Cro
ss m
ultip
ly.
sin
D�
�28s 2in 4
58°
�D
ivid
e ea
ch s
ide
by 2
4.
D�
sin
�1 �28
s 2in 458
°�
Use
the
inve
rse
sine
.
D�
81.6
°U
se a
cal
cula
tor.
58�
24
28
E
FD
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
of �
AB
C.R
oun
d a
ngl
e m
easu
res
toth
e n
eare
st d
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
1.If
c�
12,m
�A
�80
,an
d m
�C
�40
,fin
d a.
18.4
2.If
b�
20,c
�26
,an
d m
�C
�52
,fin
d m
�B
.
37
3.If
a�
18,c
�16
,an
d m
�A
�84
,fin
d m
�C
.
62
4.If
a�
25,m
�A
�72
,an
d m
�B
�17
,fin
d b.
7.7
5.If
b�
12,m
�A
�89
,an
d m
�B
�80
,fin
d a.
12.2
6.If
a�
30,c
�20
,an
d m
�A
�60
,fin
d m
�C
.
35
©G
lenc
oe/M
cGra
w-H
ill38
2G
lenc
oe G
eom
etry
Use
th
e La
w o
f Si
nes
to
So
lve
Pro
ble
ms
You
can
use
th
e L
aw o
f S
ines
to s
olve
som
e pr
oble
ms
that
in
volv
e tr
ian
gles
.
Law
of
Sin
esLe
t �
AB
Cbe
any
tria
ngle
with
a,
b, a
nd c
repr
esen
ting
the
mea
sure
s of
the
sid
es o
ppos
ite
the
angl
es w
ith m
easu
res
A,
B,
and
C,
resp
ectiv
ely.
The
n �si
n aA �
��si
n bB �
��si
n cC �
.
Isos
cele
s �
AB
Ch
as a
bas
e of
24
cen
tim
eter
s an
d a
ve
rtex
an
gle
of 6
8°.F
ind
th
e p
erim
eter
of
the
tria
ngl
e.T
he
vert
ex a
ngl
e is
68°
,so
the
sum
of
the
mea
sure
s of
th
e ba
se a
ngl
es i
s 11
2 an
d m
�A
�m
�C
�56
.
�sin b
B ��
�sin a
A�
Law
of
Sin
es
�sin 24
68°
��
�sin a56
°�
m�
B�
68,
b�
24,
m�
A�
56
asi
n 6
8°�
24 s
in 5
6°C
ross
mul
tiply
.
a�
�24si
s nin 685 °6°
�D
ivid
e ea
ch s
ide
by s
in 6
8°.
�21
.5U
se a
cal
cula
tor.
Th
e tr
ian
gle
is i
sosc
eles
,so
c�
21.5
.T
he
peri
met
er i
s 24
�21
.5 �
21.5
or
abou
t 67
cen
tim
eter
s.
Dra
w a
tri
angl
e to
go
wit
h e
ach
exe
rcis
e an
d m
ark
it
wit
h t
he
give
n i
nfo
rmat
ion
.T
hen
sol
ve t
he
pro
ble
m.R
oun
d a
ngl
e m
easu
res
to t
he
nea
rest
deg
ree
and
sid
em
easu
res
to t
he
nea
rest
ten
th.
1.O
ne
side
of
a tr
ian
gula
r ga
rden
is
42.0
fee
t.T
he
angl
es o
n e
ach
en
d of
th
is s
ide
mea
sure
66°
and
82°.
Fin
d th
e le
ngt
h o
f fe
nce
nee
ded
to e
ncl
ose
the
gard
en.
192.
9 ft
2.T
wo
rada
r st
atio
ns
Aan
d B
are
32 m
iles
apa
rt.T
hey
loc
ate
an a
irpl
ane
Xat
th
e sa
me
tim
e.T
he
thre
e po
ints
for
m �
XA
B,w
hic
h m
easu
res
46°,
and
�X
BA
,wh
ich
mea
sure
s52
°.H
ow f
ar i
s th
e ai
rpla
ne
from
eac
h s
tati
on?
25.5
mi f
rom
A;
23.2
mi f
rom
B
3.A
civ
il e
ngi
nee
r w
ants
to
dete
rmin
e th
e di
stan
ces
from
poi
nts
Aan
d B
to a
n i
nac
cess
ible
poin
t C
in a
riv
er.�
BA
Cm
easu
res
67°
and
�A
BC
mea
sure
s 52
°.If
poi
nts
Aan
d B
are
82.0
fee
t ap
art,
fin
d th
e di
stan
ce f
rom
Cto
eac
h p
oin
t.
86.3
ft
to p
oin
t B
;73
.9 f
t to
po
int
A
4.A
ran
ger
tow
er a
t po
int
Ais
42
kilo
met
ers
nor
th o
f a
ran
ger
tow
er a
t po
int
B.A
fir
e at
poin
t C
is o
bser
ved
from
bot
h t
ower
s.If
�B
AC
mea
sure
s 43
°an
d �
AB
Cm
easu
res
68°,
wh
ich
ran
ger
tow
er i
s cl
oser
to
the
fire
? H
ow m
uch
clo
ser?
Tow
er B
is 1
1 km
clo
ser
than
To
wer
A.
68� b
ca
24B
CA
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Th
e L
aw o
f S
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Th
e L
aw o
f S
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-H
ill38
3G
lenc
oe G
eom
etry
Lesson 7-6
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
from
�A
BC
.Rou
nd
an
gle
mea
sure
sto
th
e n
eare
st t
enth
deg
ree
and
sid
e m
easu
res
to t
he
nea
rest
ten
th.
1.If
m�
A�
35,m
�B
�48
,an
d b
�28
,fin
d a.
21.6
2.If
m�
B�
17,m
�C
�46
,an
d c
�18
,fin
d b.
7.3
3.If
m�
C�
86,m
�A
�51
,an
d a
�38
,fin
d c.
48.8
4.If
a�
17,b
�8,
and
m�
A�
73,f
ind
m�
B.
26.7
5.If
c�
38,b
�34
,an
d m
�B
�36
,fin
d m
�C
.41
.1 o
r 13
8.9
6.If
a�
12,c
�20
,an
d m
�C
�83
,fin
d m
�A
.36
.6
7.If
m�
A�
22,a
�18
,an
d m
�B
�10
4,fi
nd
b.46
.6
Sol
ve e
ach
�P
QR
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
8.p
�27
,q�
40,m
�P
�33
m�
Q�
53.8
,m�
R�
93.2
,r�
49.5
;o
r m
�Q
�12
6.2,
m�
R�
20.8
,r�
17.6
9.q
�12
,r�
11,m
�R
�16
m�
P�
146.
5,m
�Q
�17
.5,p
�22
.0;
or
m�
P�
1.5,
m�
Q�
162.
5,p
�1.
0
10.p
�29
,q�
34,m
�Q
�11
1m
�P
�52
.8,m
�R
�16
.2,r
�10
.2
11.I
f m
�P
�89
,p�
16,r
�12
m�
Q�
42.4
,m�
R�
48.6
,q�
10.8
12.I
f m
�Q
�10
3,m
�P
�63
,p�
13m
�R
�14
,q�
14.2
,r�
3.5
13.I
f m
�P
�96
,m�
R�
82,r
�35
m�
Q�
2,p
�35
.2,q
�1.
2
14.I
f m
�R
�49
,m�
Q�
76,r
�26
m�
P�
55,p
�28
.2,q
�33
.4
15.I
f m
�Q
�31
,m�
P�
52,p
�20
m�
R�
97,q
�13
.1,r
�25
.2
16.I
f q
�8,
m�
Q�
28,m
�R
�72
m�
P�
80,p
�16
.8,r
�16
.2
17.I
f r
�15
,p�
21,m
�P
�12
8m
�Q
�17
.7,m
�R
�34
.3,q
�8.
1
©G
lenc
oe/M
cGra
w-H
ill38
4G
lenc
oe G
eom
etry
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
from
�E
FG
.Rou
nd
an
gle
mea
sure
sto
th
e n
eare
st t
enth
deg
ree
and
sid
e m
easu
res
to t
he
nea
rest
ten
th.
1.If
m�
G�
14,m
�E
�67
,an
d e
�14
,fin
d g.
3.7
2.If
e�
12.7
,m�
E�
42,a
nd
m�
F�
61,f
ind
f.16
.6
3.If
g�
14,f
�5.
8,an
d m
�G
�83
,fin
d m
�F
.24
.3
4.If
e�
19.1
,m�
G�
34,a
nd
m�
E�
56,f
ind
g.12
.9
5.If
f�
9.6,
g�
27.4
,an
d m
�G
�43
,fin
d m
�F
.13
.8
Sol
ve e
ach
�S
TU
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
6.m
�T
�85
,s�
4.3,
t�
8.2
m�
S�
31.5
,m�
U�
63.5
,u�
7.4
7.s
�40
,u�
12,m
�S
�37
m�
T�
132.
6,m
�U
�10
.4,t
�48
.9
8.m
�U
�37
,t�
2.3,
m�
T�
17m
�S
�12
6,s
�6.
4,u
�4.
7
9.m
�S
�62
,m�
U�
59,s
�17
.8m
�T
�59
,t�
17.3
,u�
17.3
10.t
�28
.4,u
�21
.7,m
�T
�66
m�
S�
69.7
,m�
U�
44.3
,s�
29.2
11.m
�S
�89
,s�
15.3
,t�
14m
�T
�66
.2,m
�U
�24
.8,u
�6.
4
12.m
�T
�98
,m�
U�
74,u
�9.
6m
�S
�8,
s�
1.4,
t�
9.9
13.t
�11
.8,m
�S
�84
,m�
T�
47m
�U
�49
,s�
16.0
,u�
12.2
14.I
ND
IREC
T M
EASU
REM
ENT
To
fin
d th
e di
stan
ce f
rom
th
e ed
ge
of t
he
lake
to
the
tree
on
th
e is
lan
d in
th
e la
ke,H
ann
ah s
et u
p a
tria
ngu
lar
con
figu
rati
on a
s sh
own
in
th
e di
agra
m.T
he
dist
ance
fr
om l
ocat
ion
Ato
loc
atio
n B
is 8
5 m
eter
s.T
he
mea
sure
s of
th
e an
gles
at
Aan
d B
are
51°
and
83°,
resp
ecti
vely
.Wha
t is
the
dis
tanc
efr
om t
he
edge
of
the
lake
at
Bto
th
e tr
ee o
n t
he
isla
nd
at C
?
abo
ut
91.8
m
A
C
B
Pra
ctic
e (
Ave
rag
e)
Th
e L
aw o
f S
ines
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csT
he
Law
of
Sin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
©G
lenc
oe/M
cGra
w-H
ill38
5G
lenc
oe G
eom
etry
Lesson 7-6
Pre-
Act
ivit
yH
ow a
re t
rian
gles
use
d i
n r
adio
ast
ron
omy?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-6
at
the
top
of p
age
377
in y
our
text
book
.
Wh
y m
igh
t se
vera
l an
ten
nas
be
bett
er t
han
on
e si
ngl
e an
ten
na
wh
enst
udy
ing
dist
ant
obje
cts?
Sam
ple
an
swer
:O
bse
rvin
g a
n o
bje
ctfr
om
on
ly o
ne
po
siti
on
oft
en d
oes
no
t p
rovi
de
eno
ug
hin
form
atio
n t
o c
alcu
late
th
ing
s su
ch a
s th
e d
ista
nce
fro
m t
he
ob
serv
er t
o t
he
ob
ject
.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.A
ccor
din
g to
th
e L
aw o
f S
ines
,wh
ich
of
the
fo
llow
ing
are
corr
ect
stat
emen
ts?
A,F
A.� si
nmM�
�� si
nnN�
�� si
npP
�B
.�si
n Mm �
��si
Nnn
��
�sin P
p�
C.�co
msM �
��co
s nN �
��co
psP
�D
.�si
n mM �
��si
n nN �
��si
n pP
�
E.
(sin
M)2
�(s
in N
)2�
(sin
P)2
F.�si
n pP
��
�sin m
M ��
�sin n
N �
2.S
tate
wh
eth
er e
ach
of
the
foll
owin
g st
atem
ents
is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,ex
plai
n w
hy.
a.T
he
Law
of
Sin
es a
ppli
es t
o al
l tr
ian
gles
.tr
ue
b.
Th
e P
yth
agor
ean
Th
eore
m a
ppli
es t
o al
l tr
ian
gles
.Fa
lse;
sam
ple
an
swer
:It
on
ly a
pp
lies
to r
igh
t tr
ian
gle
s.c.
If y
ou a
re g
iven
th
e le
ngt
h o
f on
e si
de o
f a
tria
ngl
e an
d th
e m
easu
res
of a
ny
two
angl
es,y
ou c
an u
se t
he
Law
of
Sin
es t
o fi
nd
the
len
gth
s of
th
e ot
her
tw
o si
des.
tru
ed
.If
you
kn
ow t
he
mea
sure
s of
tw
o an
gles
of
a tr
ian
gle,
you
sh
ould
use
th
e L
aw o
f S
ines
to f
ind
the
mea
sure
of
the
thir
d an
gle.
Fals
e;sa
mp
le a
nsw
er:Y
ou
sh
ou
ld u
seth
e A
ng
le S
um
Th
eore
m.
e.A
fri
end
tell
s yo
u t
hat
in
tri
angl
e R
ST
,m�
R�
132,
r�
24 c
enti
met
ers,
and
s�
31ce
nti
met
ers.
Can
you
use
th
e L
aw o
f S
ines
to
solv
e th
e tr
ian
gle?
Exp
lain
.N
o;
sam
ple
an
swer
:In
any
tri
ang
le,t
he
lon
ges
t si
de
is o
pp
osi
te t
he
larg
est
ang
le.B
ecau
se a
tri
ang
le c
an h
ave
on
ly o
ne
ob
tuse
an
gle
,�R
mu
st b
eth
e la
rges
t an
gle
,bu
t s
�r,
so it
is im
po
ssib
le t
o h
ave
a tr
ian
gle
wit
hth
e g
iven
mea
sure
s.
Hel
pin
g Y
ou
Rem
emb
er
3.M
any
stu
den
ts r
emem
ber
mat
hem
atic
al e
quat
ion
s an
d fo
rmu
las
bett
er i
f th
ey c
an s
tate
them
in
wor
ds.S
tate
th
e L
aw o
f S
ines
in
you
r ow
n w
ords
wit
hou
t u
sin
g va
riab
les
orm
ath
emat
ical
sym
bols
.S
amp
le a
nsw
er:
In a
ny t
rian
gle
,th
e ra
tio
of
the
sin
e o
f an
an
gle
to
th
ele
ng
th o
f th
e o
pp
osi
te s
ide
is t
he
sam
e fo
r al
l th
ree
ang
les.
P
MN
p
mn
©G
lenc
oe/M
cGra
w-H
ill38
6G
lenc
oe G
eom
etry
Iden
titi
esA
n i
den
tity
is a
n e
quat
ion
th
at i
s tr
ue
for
all
valu
es o
f th
e va
riab
le f
or w
hic
h b
oth
sid
es a
re d
efin
ed.O
ne
way
to
veri
fy
an i
den
tity
is
to u
se a
rig
ht
tria
ngl
e an
d th
e de
fin
itio
ns
for
trig
onom
etri
c fu
nct
ion
s.
Ver
ify
that
(si
n A
)2�
(cos
A)2
�1
is a
n i
den
tity
.
(sin
A)2
�(c
os A
)2�
��a c� �2�
��b c� �2
��a2
� cb2
��
�c c2 2��
1
To
chec
k w
het
her
an
equ
atio
n m
aybe
an
ide
nti
ty,y
ou c
an t
est
seve
ral
valu
es.H
owev
er,s
ince
you
can
not
tes
t al
l va
lues
,you
can
not
be
cert
ain
that
th
e eq
uat
ion
is
an i
den
tity
.
Tes
t si
n 2
x�
2 si
n x
cos
xto
see
if
it c
ould
be
an i
den
tity
.
Try
x�
20.U
se a
cal
cula
tor
to e
valu
ate
each
exp
ress
ion
.
sin
2x
�si
n 4
02
sin
xco
s x
�2
(sin
20)
(cos
20)
�0.
643
�2(
0.34
2)(0
.940
)�
0.64
3
Sin
ce t
he
left
an
d ri
ght
side
s se
em e
qual
,th
e eq
uat
ion
may
be
an i
den
tity
.
Use
tri
angl
e A
BC
show
n a
bov
e.V
erif
y th
at e
ach
eq
uat
ion
is
an i
den
tity
.
1.�c so ins
AA�
�� ta
n1A
�2.
�t sa innBB
��
� co1 s
B�
�c so insAA
��
�b c�
�a c��
�b a��
� tan1
A�
�t sa innBB
��
�b a�
�b c��
�c a��
� co1 s
B�
3.ta
n B
cos
B�
sin
B4.
1�
(cos
B)2
�(s
in B
)2
tan
B c
os
B�
�b a�
�a c��
�b c��
sin
B1(
cos
B)2
�1
���a c� 2
��c c2 2�
��a c2 2�
��c2
c�2
a2�
��b c2
2 �o
r(s
in B
)2
Try
sev
eral
val
ues
for
x t
o te
st w
het
her
eac
h e
qu
atio
n c
ould
be
an i
den
tity
.
5.co
s 2x
�(c
os x
)2�
(sin
x)2
6.co
s (9
0�
x)�
sin
x
Yes;
see
stu
den
ts’w
ork
.Ye
s;se
e st
ud
ents
’wo
rk.
B
AC
ca
b
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-6
7-6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Answers (Lesson 7-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Th
e L
aw o
f C
osi
nes
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-H
ill38
7G
lenc
oe G
eom
etry
Lesson 7-7
The
Law
of
Co
sin
esA
not
her
rel
atio
nsh
ip b
etw
een
th
e si
des
and
angl
es o
f an
y tr
ian
gle
is c
alle
d th
e L
aw o
f C
osin
es.Y
ou c
an u
se t
he
Law
of
Cos
ines
if
you
kn
ow t
hre
e si
des
of a
tria
ngl
e or
if
you
kn
ow t
wo
side
s an
d th
e in
clu
ded
angl
e of
a t
rian
gle.
Let
�A
BC
be a
ny t
riang
le w
ith a
, b,
and
cre
pres
entin
g th
e m
easu
res
of t
he s
ides
opp
osite
L
aw o
f C
osi
nes
the
angl
es w
ith m
easu
res
A,
B,
and
C,
resp
ectiv
ely.
The
n th
e fo
llow
ing
equa
tions
are
tru
e.
a2
�b
2�
c2�
2bc
cos
Ab
2�
a2
�c2
�2a
cco
s B
c2�
a2
�b
2�
2ab
cos
C
In �
AB
C,f
ind
c.
c2�
a2�
b2�
2ab
cos
CLa
w o
f C
osin
es
c2�
122
�10
2�
2(12
)(10
)cos
48°
a�
12,
b�
10,
m�
C�
48
c�
�12
2�
�10
2�
�2(
12)
�(1
0)co
�s
48°
�Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
c�
9.1
Use
a c
alcu
lato
r.
In �
AB
C,f
ind
m�
A.
a2�
b2�
c2�
2bc
cos
ALa
w o
f C
osin
es
72�
52�
82�
2(5)
(8)
cos
Aa
�7,
b�
5, c
�8
49 �
25 �
64 �
80 c
os A
Mul
tiply
.
�40
��
80 c
os A
Sub
trac
t 89
fro
m e
ach
side
.
�1 2��
cos
AD
ivid
e ea
ch s
ide
by �
80.
cos�
1�1 2�
�A
Use
the
inve
rse
cosi
ne.
60°
�A
Use
a c
alcu
lato
r.
Fin
d e
ach
mea
sure
usi
ng
the
give
n m
easu
res
from
�A
BC
.Rou
nd
an
gle
mea
sure
sto
th
e n
eare
st d
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
1.If
b�
14,c
�12
,an
d m
�A
�62
,fin
d a.
13.5
2.If
a�
11,b
�10
,an
d c
�12
,fin
d m
�B
.51
3.If
a�
24,b
�18
,an
d c
�16
,fin
d m
�C
.42
4.If
a�
20,c
�25
,an
d m
�B
�82
,fin
d b.
29.8
5.If
b�
18,c
�28
,an
d m
�A
�59
,fin
d a.
24.3
6.If
a�
15,b
�19
,an
d c
�15
,fin
d m
�C
.51
58
7C
B
A
48�
1210
c
C
BA
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill38
8G
lenc
oe G
eom
etry
Use
th
e La
w o
f C
osi
nes
to
So
lve
Pro
ble
ms
You
can
use
th
e L
aw o
f C
osin
esto
solv
e so
me
prob
lem
s in
volv
ing
tria
ngl
es.
Let
�A
BC
be a
ny t
riang
le w
ith a
, b,
and
cre
pres
entin
g th
e m
easu
res
of t
he s
ides
opp
osite
the
L
aw o
f C
osi
nes
angl
es w
ith m
easu
res
A,
B,
and
C,
resp
ectiv
ely.
The
n th
e fo
llow
ing
equa
tions
are
tru
e.
a2
�b
2�
c2�
2bc
cos
Ab
2�
a2
�c2
�2a
cco
s B
c2�
a2
�b
2�
2ab
cos
C
Ms.
Jon
es w
ants
to
pu
rch
ase
a p
iece
of
lan
d w
ith
th
e sh
ape
show
n.F
ind
th
e p
erim
eter
of
the
pro
per
ty.
Use
th
e L
aw o
f C
osin
es t
o fi
nd
the
valu
e of
a.
a2�
b2�
c2�
2bc
cos
ALa
w o
f C
osin
es
a2�
3002
�20
02�
2(30
0)(2
00)
cos
88°
b�
300,
c�
200,
m�
A�
88
a�
�13
0,0
�00
��
120,
0�
00 c
os�
88°
�Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
�35
4.7
Use
a c
alcu
lato
r.
Use
th
e L
aw o
f C
osin
es a
gain
to
fin
d th
e va
lue
of c
.
c2�
a2�
b2�
2ab
cos
CLa
w o
f C
osin
es
c2�
354.
72�
3002
�2(
354.
7)(3
00)
cos
80°
a�
354.
7, b
�30
0, m
�C
�80
c�
�21
5,8
�12
.09
��
21�
2,82
0�
cos
8�
0°�Ta
ke t
he s
quar
e ro
ot o
f ea
ch s
ide.
�42
2.9
Use
a c
alcu
lato
r.
Th
e pe
rim
eter
of
the
lan
d is
300
�20
0 �
422.
9 �
200
or a
bou
t 12
23 f
eet.
Dra
w a
fig
ure
or
dia
gram
to
go w
ith
eac
h e
xerc
ise
and
mar
k i
t w
ith
th
e gi
ven
info
rmat
ion
.Th
en s
olve
th
e p
rob
lem
.Rou
nd
an
gle
mea
sure
s to
th
e n
eare
st d
egre
ean
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
1.A
tri
angu
lar
gard
en h
as d
imen
sion
s 54
fee
t,48
fee
t,an
d 62
fee
t.F
ind
the
angl
es a
t ea
chco
rner
of
the
gard
en.
75°;
48°;
57°
2.A
par
alle
logr
am h
as a
68°
angl
e an
d si
des
8 an
d 12
.Fin
d th
e le
ngt
hs
of t
he
diag
onal
s.11
.7;
16.7
3.A
n a
irpl
ane
is s
igh
ted
from
tw
o lo
cati
ons,
and
its
posi
tion
for
ms
an a
cute
tri
angl
e w
ith
them
.Th
e di
stan
ce t
o th
e ai
rpla
ne
is 2
0 m
iles
fro
m o
ne
loca
tion
wit
h a
n a
ngl
e of
elev
atio
n 4
8.0°
,an
d 40
mil
es f
rom
th
e ot
her
loc
atio
n w
ith
an
an
gle
of e
leva
tion
of
21.8
°.H
ow f
ar a
part
are
th
e tw
o lo
cati
ons?
50.5
mi
4.A
ran
ger
tow
er a
t po
int
Ais
dir
ectl
y n
orth
of
a ra
nge
r to
wer
at
poin
t B
.A f
ire
at p
oin
t C
is o
bser
ved
from
bot
h t
ower
s.T
he
dist
ance
fro
m t
he
fire
to
tow
er A
is 6
0 m
iles
,an
d th
edi
stan
ce f
rom
th
e fi
re t
o to
wer
Bis
50
mil
es.I
f m
�A
CB
�62
,fin
d th
e di
stan
ce b
etw
een
the
tow
ers.
57.3
mi
200
ft
300
ft
300
ft
88�80
�c
a
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Th
e L
aw o
f C
osi
nes
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Th
e L
aw o
f C
osi
nes
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-H
ill38
9G
lenc
oe G
eom
etry
Lesson 7-7
In �
RS
T,g
iven
th
e fo
llow
ing
mea
sure
s,fi
nd
th
e m
easu
re o
f th
e m
issi
ng
sid
e.
1.r
�5,
s�
8,m
�T
�39
t�
5.2
2.r
�6,
t�
11,m
�S
�87
s�
12.3
3.r
�9,
t�
15,m
�S
�10
3s
�19
.2
4.s
�12
,t�
10,m
�R
�58
r�
10.8
In �
HIJ
,giv
en t
he
len
gth
s of
th
e si
des
,fin
d t
he
mea
sure
of
the
stat
ed a
ngl
e to
th
en
eare
st t
enth
.
5.h
�12
,i�
18,j
�7;
m�
H24
.7
6.h
�15
,i�
16,j
�22
;m�
I46
.7
7.h
�23
,i�
27,j
�29
;m�
J70
.4
8.h
�37
,i�
21,j
�30
;m�
H91
.3
Det
erm
ine
wh
eth
er t
he
Law
of
Sin
esor
th
e L
aw o
f C
osin
essh
ould
be
use
d f
irst
to
solv
e ea
ch t
rian
gle.
Th
en s
olve
eac
h t
rian
gle.
Rou
nd
an
gle
mea
sure
s to
th
e n
eare
std
egre
e an
d s
ide
mea
sure
s to
th
e n
eare
st t
enth
.
9.10
.
Co
sin
es;
m�
A �
34;
Sin
es;
m�
L�
67;
m�
B�
80;
c�
30.7
m�
N�
27;
��
47.8
11.a
�10
,b�
14,c
�19
12.a
�12
,b�
10,m
�C
�27
Co
sin
es;
m�
A�
31;
Co
sin
es;
m�
A�
97;
m�
B�
46;
m�
C�
103
m�
B�
56;
c�
5.5
Sol
ve e
ach
�R
ST
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
13.r
�12
,s�
32,t
�34
m�
R�
20.7
,m�
S�
70.2
,m�
T�
89.1
14.r
�30
,s�
25,m
�T
�42
m�
R�
82.2
,m�
S�
55.7
,t�
20.3
15.r
�15
,s�
11,m
�R
�67
m�
S�
42.5
,m�
T�
70.5
,t�
15.4
16.r
�21
,s�
28,t
�30
m�
R�
42.3
,m�
S�
63.8
,m�
T�
74.0
M
LN
�86
�
52
24
B
AC
c
66�
33
19
©G
lenc
oe/M
cGra
w-H
ill39
0G
lenc
oe G
eom
etry
In �
JK
L,g
iven
th
e fo
llow
ing
mea
sure
s,fi
nd
th
e m
easu
re o
f th
e m
issi
ng
sid
e.
1.j
�1.
3,k
�10
,m�
L�
77�
�9.
8
2.j
�9.
6,�
�1.
7,m
�K
�43
k�
8.4
3.j
�11
,k�
7,m
�L
�63
��
10.0
4.k
�4.
7,�
�5.
2,m
�J
�11
2j�
8.2
In �
MN
Q,g
iven
th
e le
ngt
hs
of t
he
sid
es,f
ind
th
e m
easu
re o
f th
e st
ated
an
gle
toth
e n
eare
st t
enth
.
5.m
�17
,n�
23,q
�25
;m�
Q75
.7
6.m
�24
,n�
28,q
�34
;m�
M44
.2
7.m
�12
.9,n
�18
,q�
20.5
;m�
N60
.2
8.m
�23
,n�
30.1
,q�
42;m
�Q
103.
7
Det
erm
ine
wh
eth
er t
he
Law
of
Sin
es o
r th
e L
aw o
f C
osin
es s
hou
ld b
e u
sed
fir
st t
oso
lve
�A
BC
.Th
en s
ole
each
tri
angl
e.R
oun
d a
ngl
e m
easu
res
to t
he
nea
rest
deg
ree
and
sid
e m
easu
re t
o th
e n
eare
st t
enth
.
9.a
�13
,b�
18,c
�19
10.a
�6,
b�
19,m
�C
�38
Co
sin
es;
m�
A�
41;
Co
sin
es;
m�
A�
15;
m�
B�
65;
m�
C�
74m
�B
�12
7;c
�14
.7
11.a
�17
,b�
22,m
�B
�49
12.a
�15
.5,b
�18
,m�
C�
72
Sin
es;
m�
A�
36;
Co
sin
es;
m�
A�
48;
m�
C�
95;
c�
29.0
m�
B�
60;
c�
19.8
Sol
ve e
ach
�F
GH
des
crib
ed b
elow
.Rou
nd
mea
sure
s to
th
e n
eare
st t
enth
.
13.m
�F
�54
,f�
12.5
,g�
11m
�G
�45
.4,m
�H
�80
.6,h
�15
.2
14.f
�20
,g�
23,m
�H
�47
m�
F�
57.4
,m�
G �
75.6
,h�
17.4
15.f
�15
.8,g
�11
,h�
14m
�F
�77
.4,m
�G
�42
.8,m
�H
�59
.8
16.f
�36
,h�
30,m
�G
�54
m�
F�
73.1
,m�
H�
52.9
,g�
30.4
17.R
EAL
ESTA
TET
he
Esp
osit
o fa
mil
y pu
rch
ased
a t
rian
gula
r pl
ot o
f la
nd
on w
hic
h t
hey
plan
to
buil
d a
barn
an
d co
rral
.Th
e le
ngt
hs
of t
he
side
s of
th
e pl
ot a
re 3
20 f
eet,
286
feet
,an
d 30
5 fe
et.W
hat
are
th
e m
easu
res
of t
he
angl
es f
orm
ed o
n e
ach
sid
e of
th
e pr
oper
ty?
65.5
,54.
4,60
.1
Pra
ctic
e (
Ave
rag
e)
Th
e L
aw o
f C
osi
nes
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
Answers (Lesson 7-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csT
he
Law
of
Co
sin
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
7-7
7-7
©G
lenc
oe/M
cGra
w-H
ill39
1G
lenc
oe G
eom
etry
Lesson 7-7
Pre-
Act
ivit
yH
ow a
re t
rian
gles
use
d i
n b
uil
din
g d
esig
n?
Rea
d th
e in
trod
uct
ion
to
Les
son
7-7
at
the
top
of p
age
385
in y
our
text
book
.
Wh
at c
ould
be
a di
sadv
anta
ge o
f a
tria
ngu
lar
room
?S
amp
le a
nsw
er:
Fu
rnit
ure
will
no
t fi
t in
th
e co
rner
s.
Rea
din
g t
he
Less
on
1.R
efer
to
the
figu
re.A
ccor
din
g to
th
e L
aw o
f C
osin
es,w
hic
h
stat
emen
ts a
re c
orre
ct f
or �
DE
F?
B,E
,HA
.d
2�
e2�
f2�
efco
s D
B.e
2�
d2
�f2
�2d
fco
s E
C.
d2
�e2
�f2
�2e
fco
s D
D.f
2�
d2
�e2
�2e
fco
s F
E.
f2�
d2
�e2
�2d
eco
s F
F.d
2�
e2�
f2
G.�
sin d
D ��
�sin e
E ��
�sin f
F�
H.d
��
e2�
f�
2�
2e�
fco
s �
D�
2.E
ach
of
the
foll
owin
g de
scri
bes
thre
e gi
ven
par
ts o
f a
tria
ngl
e.In
eac
h c
ase,
indi
cate
wh
eth
er y
ou w
ould
use
th
e L
aw o
f S
ines
or
the
Law
of
Cos
ines
fir
st i
n s
olvi
ng
a tr
ian
gle
wit
h t
hos
e gi
ven
par
ts.(
In s
ome
case
s,on
ly o
ne
of t
he
two
law
s w
ould
be
use
d in
sol
vin
gth
e tr
ian
gle.
)
a.S
SS
Law
of
Co
sin
esb
.AS
A L
aw o
f S
ines
c.A
AS
Law
of
Sin
esd
.SA
S L
aw o
f C
osi
nes
e.S
SA
Law
of
Sin
es
3.In
dica
te w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,exp
lain
wh
y.
a.T
he
Law
of
Cos
ines
app
lies
to
righ
t tr
ian
gles
. tru
eb
.T
he
Pyt
hag
orea
n T
heo
rem
app
lies
to
acu
te t
rian
gles
.Fal
se;
sam
ple
an
swer
:It
on
ly a
pp
lies
to r
igh
t tr
ian
gle
s.c.
Th
e L
aw o
f C
osin
es i
s u
sed
to f
ind
the
thir
d si
de o
f a
tria
ngl
e w
hen
you
are
giv
en t
he
mea
sure
s of
tw
o si
des
and
the
non
incl
ude
d an
gle.
Fals
e;sa
mp
le a
nsw
er:
It is
use
d w
hen
yo
u a
re g
iven
th
e m
easu
res
of
two
sid
es a
nd
th
e in
clu
ded
ang
le.
d.
Th
e L
aw o
f C
osin
es c
an b
e u
sed
to s
olve
a t
rian
gle
in w
hic
h t
he
mea
sure
s of
th
e th
ree
side
s ar
e 5
cen
tim
eter
s,8
cen
tim
eter
s,an
d 15
cen
tim
eter
s.Fa
lse;
sam
ple
answ
er:
5 �
8 �
15,s
o,b
y th
e Tr
ian
gle
Ineq
ual
ity
Th
eore
m,n
o s
uch
tria
ng
le e
xist
s.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al f
orm
ula
is
to r
elat
e it
to
one
you
alr
eady
know
.Th
e L
aw o
f C
osin
es l
ooks
som
ewh
at l
ike
the
Pyt
hag
orea
n T
heo
rem
.Bot
h f
orm
ula
sm
ust
be
tru
e fo
r a
righ
t tr
ian
gle.
How
can
th
at b
e? c
os
90 �
0,so
in a
rig
ht
tria
ng
le,w
her
e th
e in
clu
ded
an
gle
is t
he
rig
ht
ang
le,t
he
Law
of
Co
sin
esb
eco
mes
th
e P
yth
ago
rean
Th
eore
m.
D
dE
e F
f
©G
lenc
oe/M
cGra
w-H
ill39
2G
lenc
oe G
eom
etry
Sp
her
ical
Tri
ang
les
Sph
eric
al t
rigo
nom
etry
is
an e
xten
sion
of
plan
e tr
igon
omet
ry.
Fig
ure
s ar
e dr
awn
on
th
e su
rfac
e of
a s
pher
e.A
rcs
of g
reat
ci
rcle
s co
rres
pon
d to
lin
e se
gmen
ts i
n t
he
plan
e.T
he
arcs
of
thre
e gr
eat
circ
les
inte
rsec
tin
g on
a s
pher
e fo
rm a
sph
eric
al
tria
ngl
e.A
ngl
es h
ave
the
sam
e m
easu
re a
s th
e ta
nge
nt
lin
es
draw
n t
o ea
ch g
reat
cir
cle
at t
he
vert
ex.
Sin
ce t
he
side
s ar
e ar
cs,t
hey
too
can
be
mea
sure
d in
deg
rees
.
Sol
ve t
he
sph
eric
al t
rian
gle
give
n a
�72
�,b
�10
5�,a
nd
c�
61�.
Use
th
e L
aw o
f C
osin
es.
0.30
90�
(–0
.258
8)(0
.484
8)�
(0.9
659)
(0.8
746)
cos
Aco
s A
�0.
5143
A�
59°
�0.
2588
� (
0.30
90)(
0.48
48)�
(0.9
511)
(0.8
746)
cos
Bco
s B
��
0.49
12B
�11
9°
0.48
48�
(0.
3090
)(–0
.258
8)�
(0.9
511)
(0.9
659)
cos
Cco
s C
�0.
6148
C�
52°
Ch
eck
by u
sin
g th
e L
aw o
f S
ines
.
�s si in n7 52 9° °�
��s si in n
1 10 15 9° °�
��s si in n
6 51 2° °�
�1.
1
Sol
ve e
ach
sp
her
ical
tri
angl
e.
1.a
�56
°,b
�53
°,c
�94
°2.
a�
110°
,b�
33°,
c�
97°
A�
41�,
B�
39�,
C�
128�
A�
116�
,B�
31�,
C�
71�
3.a
�76
°,b
�11
0°,C
�49
°4.
b�
94°,
c�
55°,
A�
48°
A�
59�,
B�
124�
,c�
59�
a�
60�,
B�
121�
,C�
45�
A
C
B
c
ba
En
rich
men
t
NA
ME
____
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7-7
7-7
The
sum
of
the
side
s of
a s
pher
ical
tria
ngle
is le
ss t
han
360°
.T
he s
um o
f th
e an
gles
is g
reat
er t
han
180°
and
less
tha
n 54
0°.
The
Law
of
Sin
es f
or s
pher
ical
tria
ngle
s is
as
follo
ws.
� ss ii nnAa
��
� ss ii nnBb
��
� ss ii nnCc
�
The
re is
als
o a
Law
of
Cos
ines
for
sph
eric
al t
riang
les.
cos
a�
cos
bco
s c
�si
n b
sin
cco
s A
cos
b�
cos
aco
s c
�si
n a
sin
cco
s B
cos
c�
cos
aco
s b
�si
n a
sin
bco
s C
Exam
ple
Exam
ple
Answers (Lesson 7-7)