chapter 10: right triangles and circles

T H E M E : Architecture

Imagine that you have the opportunity to design and build a new home foryour family. How would you decide what design features to incorporate?

Your new home must be functional to meet your family’s needs. It must alsobe appealing to the eye to suit your family’s personality and complementother buildings in the community.

Architects have been using geometric principles for centuries to design andbuild homes, schools and public buildings. In the search for pleasing and usefulforms, they have explored and applied many important principles of geometry.

• Building Inspectors (page 435) ensure the safety of buildings beforeconstruction is complete. They make sure that safety guidelines andbuilding codes are strictly followed. Inspectors may stop a project ifsafety standards are not met.

• Landscape Architects(page 453) design outdoor areassuch as gardens, parks, andplaygrounds. Landscapearchitects study soil, sunlight,topography, and climate whendesigning a landscaping plan.

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Right Trianglesand Circles

CH

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Chapter 10 Right Triangles and Circles

Use the table for Questions 1–4.

1. Compare the number of stories to the height in feet for eachbuilding. Which building has the least amount of height perstory? What has the greatest amount of height per story?

2. The framework for the Empire State Building rose at a rate offour-and-a-half stories per week. How many days were requiredto build the framework?

3. The Sears Tower consists of nine 75-ft square modules which riseto staggered levels. What is the combined area of the modules insquare feet?

4. Create a bar graph of the buildings’ heights with the buildingarranged according to the date they were built.

CHAPTER INVESTIGATIONOften, large cities do not have available space to build newbuildings or the means to tear down older buildings and replacethem with new ones. As a result, many cities are hiring architects togive older buildings a facelift or makeover. A new facade is attachedto the front of the building to make the building more attractive andunify design elements.

Working TogetherMake a sketch of the front of your school as it now looks. Researcharchitectural styles and select appropriate design elements. Thencreate a facade that could fit over the front of your school to give it anew look. Draw plans for your new design. Use the ChapterInvestigation icons to guide your group.

Data Activity: Tall Buildings

423

Tall Buildings Skyscraper

Central Towers

Petronas Towers

Sears Tower

Jin Mao Building

Empire State Building

Bank of China

Hong Kong

Kuala Lumpur

Chicago

Shanghai

New York

Hong Kong

1992

1997

1974

1999

1931

1989

City Built

1227 ft

1483 ft

1450 ft

1379 ft

1250 ft

1209 ft

Height

78

88

110

88

102

70

Stories

Least—Empire StateBuilding, about 12.3 ft per story.Greatest—Bank ofChina, about 17.3 ftper story.

about 159 days

50,625 ft2

Check students’ work.

The skills on these two pages are ones you have already learned. Review theexamples and complete the exercises. For additional practice on these and moreprerequisite skills, see pages 654–661.

PERCENT OF A NUMBER

Example Find 42% of 624.Change the percent to a decimal: 42% � 0.42Multiply: 624

� 0.42262.08

42% of 624 is 262.08.

Find the given percent of each number.

1. 15% of 500 2. 50% of 768 3. 24% of 496

4. 33% of 127 5. 64% of 481 6. 93% of 722

7. 81% of 3297 8. 29.6% of 17.84 9. 47% of 82.56

10. 23.5% of 1604 11. 106% of 300 12. 17.8% of 296

CIRCUMFERENCE AND AREA

The formula for the circumference of a circle is C � 2�r, where r is the radius ofthe circle. The formula for the area of a circle is A � �r 2, where r is the radius ofthe circle.

Find the circumference and area of each circle. Use 3.14 for �. Round answersto the nearest hundredth if necessary.

13. 14. 15.

16. 17. 18.

5 cm

6.8 in.17 cm

1.8 cm

5.2 in.3 cm


10CH

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10 Are You Ready?Refresh Your Math Skills for Chapter 10

75

41.91

2670.57

376.94

384

307.84

5.28064

318

119.04

671.46

38.8032

52.688

C � 11.30 cm;A � 10.17 cm2

C � 32.66 in.;A � 84.91 in.2

C � 18.84 cm;A � 28.26 cm2

C � 53.58 cm;A � 226.87 cm2

C � 21.35 in.;A � 36.30 in.2

C � 15.7 cm;A � 19.63 cm2

Chapter 10 Are You Ready?

SQUARES AND SQUARE ROOTS

Simplify each expression.

19. 162 20. �144� 21. 242 22. �64�

23. 92 24. �529� 25. 152 26. �196�

27. 132 28. �289� 29. 302 30. �100�

ANGLES

Find the measure of each indicated angle.

31. �AEB

32. �AED

33. �DEC

34. �DCF

35. �EFB

36. �HFC

37. �FEB

38. �ABG

39. �CBE

40. �BAD

41. �FAE

42. �EAD

43. �CAE

44. �CAF

45. �DAF

B

C

D

E

F

A40°

115°

A

D

H

G

E(2x � 7)° (3x � 2)°

F

BC

AC || DF

A

D

B

C

E49°

A D

H

G

E

40°

F

B C

256

81

169

12

23

17

576

225

900

8

14

10

131°

49°

131°

140°

130°

130°

77°

103°

103°

130°

90°

65°

155°

50°

118°

425

Work with a partner to explore square roots.

A calculator gives the square root of 2 as 1.414213562. Copy and complete this chart, using a calculator.

(1.4)2 � ___?__ (1.4142)2 � ___?__ (1.4142135)2 � ___?__

(1.41)2 � ___?__ (1.41421)2 � ___?__ (1.41421356)2 � ___?__

(1.414)2 � ___?__ (1.414213)2 � ___?__ (1.414213562)2 � ___?__

Do you think that eventually you will find a number, y, with enough decimal places so that y 2 � 2 exactly?

BUILD UNDERSTANDING

The symbol �x� means the square root of a number, x. It is the number thatmultiplied by itself equals x. For example, �4� � 2, because 22 � 4; and �144� � 12, because 122 � 144. These two examples have square roots that arerational numbers.

The number �2� is an irrational number, which means that it is not a rationalnumber. Therefore, it is a number that cannot be written as a fraction, aterminating decimal, or a repeating decimal. It can be represented by anonterminating, nonrepeating decimal, and it can be approximated to anydecimal place.

E x a m p l e 1

ARCHITECTURE An architect is drawing a landscaping planwhich includes three square cement blocks with areas of thebases approximately 5, 31, and 98 ft2. To find the length of aside of each block, she finds the square root of each area.Find the value of each to the nearest hundredth: �5�, �31�and �98�.

SolutionUse a calculator. Then round to the hundredths place.

�5� � 2.236067977 . . . rounds to 2.24

�31� � 5.567764363 . . . rounds to 5.57

�98� � 9.899494937 . . . rounds to 9.90

Chapter 10 Right Triangles and Circles426

10-1 IrrationalNumbersGoals ■ Find square roots.

■ Simplify products and quotients containing radicals.

Applications Architecture, Small Business, Construction

TechnologyNote

To find the square root of2, the key sequence for most scientific calculatorsis 2 .

For graphing calculators,the key sequence is usually

2 .ENTER

1.96 1.999999824

1.999999993

1.999999999

1.999998409

1.9999899241.9881

1.999396

no

1.99996164

Another way to read the expression �2� is “radical 2.” The number under theradical sign is called the radicand. There are properties of radicals that allow youto simplify them.

E x a m p l e 2

Simplify each expression.

a. �32� b. �300� c. �125�

SolutionRewrite each radicand as a product of two numbers, so that one of them is aperfect square.

a. �32� � �16 � 2� � �16� � �2� or 4�2�

b. �300� � �3 � 10�0� � �3� � �100� or 10�3�

c. �125� � �25 � 5� � �25� � �5� or 5�5�

Numbers written in radical form can be multiplied together.

E x a m p l e 3

Multiply (4�10�)(� 3.1�6�).

SolutionRewrite the product so that rational factors and irrational factors are grouped separately.

(4�10�)(�3.1�6�) � (4 � �3.1)(�10� � �6�)

� �12.4�60� �10� � �6� � �10 � 6�

The product can be simplified.

�12.4�60� � �12.4(�4 � 15�)

� �12.4(2�15�)

� �24.8�15�

So, (4�10�)(�3.1�6�) � �24.8�15�.

A similar theorem about radicals applies to quotients.

Lesson 10-1 Irrational Numbers 427

The square root of a quotient of two nonnegative numbers isequal to the quotient of their square roots.

��ab

�� where b � 0�a��b�

Theorem

The square root of the product of two nonnegative numbers isthe same as the product of their square roots.

�a � b� � �a� � �b�Theorem

CheckUnderstanding

What is the product, insimplest radical form, of�14� � �7�?

7�2�

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E x a m p l e 4

Express the quotient for �5�33� � 3�22� in simplest radical form.

Solution� � �

�

35� � ��

32

32�� Simplify �

32

32�.

� ��

35� � ��

32

��

In simplest radical form, denominators cannot include radicals.

� Multiply numerator and denominator by �2�.

�

� , or ��56�6��

The process of rewriting a quotient to eliminate radicals from the denominator is called rationalizing the denominator.

TRY THESE EXERCISES

CALCULATOR Find each value to the nearest hundredth.

1. �11� 2. �56� 3. �85� 4. �196�

Write each square root in simplest radical form.

5. �44� 6. �242� 7. �75� 8. �48�

Simplify.

9. (2�6�) (7�2�) 10. (�3�10�) (5�5�) 11. (4�2�)2

12. 13. 14. ��221��

PRACTICE EXERCISES

CALCULATOR Find each value to the nearest hundredth.

15. �21� 16. �47� 17. �73� 18. �200�

Write each expression in simplest radical form.

19. �162� 20. �500� 21. �72�

22. �40� 23. (2�3�) (4�6�) 24. (10�2�) (2�6�)

25. (3�3�)2 26. 27. ��7

2��

28. ��65

�� 29. (2�5�)2 30.3�12��4�48�

�45��5�

2��3�

�150��

�6�

�5�6��

3 � 2

�5�6��

3�4�

�5�3�(�2�)��

3�2�(�2�)�5�3��

3�2�

�5�3��

3�2�

5�33��3�22�


Reading Math

In everyday language,words that have“rational” as one of theirparts usually relate toreasons or reasonableness.For example, rationalizingmeans to find a reason forsomething. When you saysomeone is beingirrational, you mean thathe or she is not makingsense or not beingreasonable.

In mathematics, wordswith “rational” as one oftheir parts always relateto ratio. Rational numbersare numbers that can bewritten as a ratio of twointegers. Irrationalnumbers cannot.Rationalizing thedenominator meanschanging it to a rationalnumber.

3.32 7.48 9.22 14.00

4�3�5�3�11�2�2�11�

28�3�

5 �2�

33�

�

�75�2� 32

��

242��

14.148.546.864.58

9�2�

2�10�

27

��

530�� 20

10�5�

24�2�

3

�38

�

�7�

22�

�

40�3�

6�2�

PRACTICE EXERCISES • For Extra Practice, see page 693.

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Lesson 10-1 Irrational Numbers 429

31. SMALL BUSINESS Georgia is starting apet-sitting service in her backyard. Shewants to enclose an area of about 6 m2,and has decided that the best shape forthe enclosure is a square. What shouldbe the length of a side of the enclosure,to the nearest tenth of a meter?

32. WRITING MATH Is the number�207.36� rational or irrational? Explain.

33. DATA FILE Use the data on rectangulararchitectural structures on page 645. Forthe Bakong Temple and the Wat KukatTemple, express the length of a side asthe square root of the area of thestructure in meters.

34. CONSTRUCTION A mason is building a square pedestal from brick. The topof the pedestal must have an area of about 12 ft2. To the nearest hundredth,find the length of a side of the pedestal.

The expression �n x� means the nth root of x. It means the number which, whenraised to the nth power, equals x. Example: �3 8� � 2, because 23 � 8.

Find each of the following.

35. �3 27� 36. �3 125� 37. �4 16� 38. �4 81�

MATH HISTORY In the first century A.D., Heron of Alexandria discovered aformula for finding the area of a triangle when only the measures of its sides areknown:

A � �s(s � a�)(s � b�)(s � c�)� where s � 0.5(a � b � c).

Use Heron’s formula to find the area of each triangle to the nearest tenth.

39. 3 cm, 5 cm, 6 cm 40. 7 in., 12 in., 15 in.

EXTENDED PRACTICE EXERCISES

Decide if each statement is true or false. If it is false, give a counterexample.

41. The product of two even numbers is always an even number.

42. The square of a rational number is always a rational number.

43. The product of two irrational numbers is always an irrational number.

44. CHAPTER INVESTIGATION Make a detailed sketch of the front of the mainbuilding at your school. Include any current architectural elements. Estimatemeasurements for the elements in the sketch. If possible, consult buildingplans to find the length and width of standard doors and windows.

MIXED REVIEW EXERCISES

Change each unit of measure as indicated. (Lesson 5-1)

45. 8 c � ___?__ qt 46. 7 yd � ___?__ in. 47. 10 km � ___?__ m

48. 6 gal � ___?__ pt 49. 14 m � ___?__ dm 50. 7 g � ___?__ kg

2.4 m

3.46 ft

3 5 2 3

7.5 cm2 41.2 in.2

true

true

2

48 140

252 10,000

0.007

false; possible counterexample (�2�)(�18�) � 6

Bakong Temple, �4900� m; Wat Kuhat Temple, �529� m

32. �207.3�6�is rational,since14.42 �207.36.


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Draw any right triangle on graph paper. Make squares on each side ofthe triangle, as shown. Cut out the squares. Try to fit the two smallersquares on top of the larger square. You may cut up the smaller squares.What conclusion can you draw?

BUILD UNDERSTANDING

In a right triangle, the shorter sides are legs, and the side opposite the rightangle is the hypotenuse. The relationship between the squares of the lengths of thelegs and the square of the length of the hypotenuse is called the Pythagorean Theorem.

This proof of the Pythagorean Theorem usessimilar triangles formed by the altitude to thehypotenuse.

Given �ABC, �C is a right angle.Prove a 2 � b 2 � c 2

Statements Reasons

1. �ABC is a right triangle. 1. Given2. C�D�is perpendicular to AB. 2. There is one and only one line

through a point perpendicular3. C�D�is the altitude to the to a given line.

hypotenuse in �ABC. 3. Definition of altitude.4. �ABC � �ACD � �CBD 4. The altitude to the hypotenuse

forms two right triangles thatare similar to each other andto the original triangle.

5. �ac

� � �ax

�, �bc

� � �by

� 5. Corresponding sides of similartriangles are proportional.

6. cx � a 2, cy � b 2 6. Cross products are equal.7. cx � cy � a 2 � b 2 7. Addition property of equality8. c(x � y) � a 2 � b 2 8. Distributive property9. c � x � y 9. Segment addition postulate

10. c(c) � a 2 � b 2 10. Substitution11. c 2 � a 2 � b 2 11. Definition of c 2.


10-2 The PythagoreanTheoremGoals ■ Use the Pythagorean Theorem to solve problems

involving right triangles.

Applications Architecture, Construction, Home Repairs

a

b x

A

C B

Dy

c

In a right triangle, the sum of the squares ofthe measures of the two legs is equal to thesquare of the measure of the hypotenuse.

a 2 � b 2 � c 2

PythagoreanTheorem

a

b

c

Math: Who,Where, When

The scarecrow in theclassic movie TheWizard of Oz tried torecite the PythagoreanTheorem as proof ofintelligence when hereceived his “brains.”But he stated itincorrectly.

The sum of the squares of the lengths ofthe two shorter sides of a right triangleequals the square of the length of thelongest side.

You can use the Pythagorean Theorem to find the length of a side of a righttriangle when the lengths of the other two sides are known.

E x a m p l e 1

ARCHITECTURE An architect draws a right triangle, �DEF, on a blueprint. DE � 3 cm and EF � 5 cm.

Find DF to the nearest tenth.

SolutionLet x � DF. 32 � 52 � x 2

9 � 25 � x 2

34 � x 2

�34� � x5.830951895 . . . � x Use a calculator or a square root table to find �34�.

So, DF � 5.8 cm to the nearest tenth.

GEOMETRY SOFTWARE You can use geometry software to solve problemsinvolving right triangles. Set the distance units to either inches, centimeters, or pixels. Choose pixels when the problem involves large numbers. Use the Show Grid option to make drawing figures easier. Draw the lengths given in theproblem. Use the software to check the measurement of these segments. Drawand measure any remaining segments. Use the software to find unknown lengthsand angle measures as required by the problem.

Sometimes, the missing measure in the right triangle is the length of one of the legs.

E x a m p l e 2

Find x to the nearest hundredth of an inch.

Solutionx 2 � 92 � 112

x 2 � 81 � 121x 2 � 40

x � �40�x � 6.32455532 . . . So, x � 6.32 in.

The converse of the Pythagorean Theorem is also true.

E x a m p l e 3

Are triangles with the following side lengths right triangles?

a. 2 cm, 4�2� cm, 6 cm b. 4 in., 9 in., 10 in.

Lesson 10-2 The Pythagorean Theorem 431

D

E F

3 cm

5 cm

9 in.

11 in.x

If the sum of the squares of the measures of twoshorter sides of a triangle is equal to the squareof the measure of the third side, then thetriangle is a right triangle.

Converse of thePythagorean

Theorem





Solutiona. 22 � (4�2�)2 � 62 b. 42 � 92 � 102

4 � (16 � 2) � 36 16 � 81 � 100

4 � 32 � 36 97 � 100

36 � 36

Therefore, three sides with lengths 2 cm, 4�2� cm, and 6 cm do form a righttriangle, but sides with lengths 4 in., 9 in., and 10 in. do not.

TRY THESE EXERCISES

Use the Pythagorean Theorem to find the unknown length. Round youranswers to the nearest tenth.

1. 2. 3.

4. CONSTRUCTION A triangular brace has sides that measure 6 cm, 8 cm, and10 cm. Does the brace have a 90° angle?

5. HOME REPAIR Tim is cleaning out the rain gutters on his home. He has an18-ft ladder. If the base of the ladder is placed 5 ft from the base of thebuilding, how far up the wall will the ladder reach?

PRACTICE EXERCISES

Use the Pythagorean Theorem to find the unknown length. Round youranswers to the nearest tenth.

6. 7. 8.

9. 10. 11.

Determine if the triangle is a right triangle. Write yes or no.

12. 13. 14.

8

10

13

8

15

17

9

12

15

8 ft

8 ft

10 m18 m9 cm

12 cm

6 in.10 in.5 cm

8 cm

7 m

24 m

5 ft 7 ft

815 cm7 m

7 m


9.9 m

12.7 cm8.6 ft

yes

about 17.3 ft

25 m9.4 cm

8 in.

11.3 ft15.0 m7.9 cm

yes yesno


Solve. Round your answers to the nearest tenth.

15. GEOMETRY SOFTWARE Find the length of the diagonal of arectangle with a length of 5 cm and a width of 4 cm.

16. A pole 4 m high is to be attached by a guy wire to a stakein the ground 1.2 m from the base of the pole. How long must the guy wire be?

17. A ramp 6 yd long reaches from the loading dock to a point onthe ground 4 yd from the base of the dock. How high aboveground is the loading dock?

Find x in each figure. Round your answer to the nearest tenth.

18. 19. 20.

21. TALK ABOUT IT Using s for the length of the side of a square and d for thelength of the diagonal of the square, Michi has written the formula d � s�2�.She says that the formula can be used to find the length of the diagonal ofany square if the length of a side is known. Do you agree?

22. Figure ABCDEFGH is a rectangular prism. Its length AB is 8 cm, its width BC is 6 cm, and its height BF is 24 cm. Find the length of the diagonal B

⎯H⎯

. (Hint: �BDH is a right triangle.)


23. Prove that, in any rectangular prism with length l, width w, and height h, thelength d of a diagonal is given by the formula, d � �l 2 � w�2 � h 2�.

24. Three positive integers, a, b, and c, for which a 2 � b 2 � c 2 form a Pythagorean triple. Let x and y represent two positive integers such that x � y.Let a � 2xy, b � x 2 � y 2, and c � x 2 � y 2. Try three different pairs of valuesfor x and y. What do you notice?

25. WRITING MATH In ancient Egypt, a pair of workers called rope stretcherswould use a loop of rope divided by knots into 12 equal parts to mark off aperfect right angle in the sand. They would drive a stake through one knot.One worker would pull the rope taut at the third knot from the stake, whilethe other pulled the rope taut at the fourth knot on the other side of thestake. Write a paragraph and draw a diagram explaining why this systemworked.


A die is rolled 200 times with the following results: (Lesson 9-1)Outcome 1 2 3 4 5 6Frequency 31 40 30 29 34 36

What is the experimental probability of rolling each of the following results?

26. 3 27. 5 28. 6 29. 1 30. a number less than 5

14 cm x cm

x cm

3 in.

6 in.

x in.

1 m

1 m

1 m

x m

Lesson 10-2 The Pythagorean Theorem 433

A B

CD

E F

GH

6.4 cm

about 4.2 m

about 4.5 yd

1.7 6.7

9.9

Yes

26 cm

See additional answers.

The values of a, b, and c form a Pythagoreantriple.


0.15 0.17 0.18 0.155 0.65



PRACTICE LESSON 10-1Find each value to the nearest hundredth.

1. �53� 2. �150� 3. 2�7� 4. �624�

5. 10�10� 6. �245� 7. �1000� 8. �37�

Write each in simplest radical form.

9. �50� 10. �48� 11. �72� 12. �600�

13. �80� 14. �192� 15. �88� 16. �147�

17. �242� 18. 3�20� 19. 5�63� 20. 10�75�

21. �2�3��2

22. 23. ��34

�� 24. �3�2��5�10��25. 26. �8�18��4�2�� 27. ��

176�� 28. ��

38

�� 2�27��

Find each of the following.

29. �3 64� 30. �4 625� 31. �5 32� 32. �5 243�

33. �3 1000� 34. �3 �18

�� 35. �3 �217�� 36. �3

�8�

PRACTICE LESSON 10-2Use the Pythagorean Theorem to find the missing length. Round answers to the nearest tenth.

37. 38. 39.

40. 41. 42.

Solve. Round your answers to the nearest tenth.

43. Find the length of the diagonal of a rectangle with a length of 18 cm and awidth of 24 cm.

44. Find the width of a rectangle with a length of 30 ft and a diagonal of 34 ft.

51 m24 m

52 yd39 yd

47 cm21 cm

11 in.

8 in.55 ft

44 ft

24 m

10 m

3��2�

�10��2�

�12��3�


7.28

31.62

12.25

15.65

5.29

31.62

24.98

6.08

5�2�

4�5�

11�2�

12

�5�

4

10

4�3�

8�3�

6�5�

2

192

5

�12

�

6�2�

2�22�

15�7�

��23�

�

�4�

77�

�

2

�13

�

10�6�

7�3�

50�3�

30�5�

13.5

3

�2

13.6 in.33 ft26 m

42.0 cm65 yd 45 m

30 cm

16 ft

Review and Practice Your Skills

PRACTICE LESSON 10-1–LESSON 10-2Write each in simplest radical form. (Lesson 10-1)

45. �24� 46. �28� 47. �40� 48. �52�

49. ��15

�� 50. 51. 52. ��168��

53. 8�63� 54. 5�200� 55. 11�96� 56. �2�72�

57. ��24��3�6�� 58. �2�5��5�20�� 59. �7�14��3�2��8�21�� 60. �x 2�

Determine if the triangle is a right triangle. Write yes or no. (Lesson 10-2)

61. 62. 63. 42

25.233.6

60

65

25

14

3�5

�28��7�

4��2�

2�6�

��55�

�

24�7�

36100

2�7�

2�2�

50�2�

2�10�

2

44�6�

2352�3�

x

2�13�

��33�

�

�12�2�

yes

yes

no

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Workplace Knowhow

Career – Construction and Building Inspectors

Construction and building inspectors make sure new structures follow building codes and ordinances, zoning

regulations, and contract specifications. They may also inspectalterations and repairs to existing structures.

Inspectors have the authority to stop projects if safety standardsand building codes are not met. To do their work, inspectors usetape measures, survey instruments, metering devices, cameras,and calculators.

At a construction site, a temporary ramp hasbeen built to assist workers in transportingmaterials up to the floor level. A diagram ofthe ramp is shown.

1. Use the Pythagorean Theorem to find thelength of the ramp to the nearest tenth.

2. What is the measure of �EFD?

3. What is the slope of the ramp?

4. If DF is lengthened to 30 ft and CE remains the same, what will the newmeasure of EF be (to the nearest tenth)?

19.7 ft24°

�49

�

27.2 ft

4 ft

8 ft

22 ft

156°

C E

FD

Chapter 10 Review and Practice Your Skills 435

http://mathmatters3.com/mathworks

Work with a partner. You will need a compass, straightedge, and scissors.

1. Construct several different equilateral triangles.

2. Cut out each triangle and fold it so that one vertex matches another.

3. Draw a line segment along the fold.

4. Examine the figures formed by the triangle and the line segment you drew.Write as many different observations about the triangle and the line segmentas you can.

BUILD UNDERSTANDING

All right triangles with the same corresponding angle measures are similar. Thisfact and the Pythagorean Theorem lead to some theorems about therelationships among the sides in two types of right triangles.

In the equilateral triangles you examined above, you probably noticed that thesegment that separated one side into two congruent segments was alsoperpendicular to that segment. In this way, a right triangle is formed that has a60° angle and one leg that is half the measure of the hypotenuse.

E x a m p l e 1

ARCHITECTURE A triangular support shown on a blueprint forms a 30°–60°–90°triangle. On the plans, the leg opposite the 30° angle measures 4 cm. What arethe measures of the other two sides?


10-3 Special RightTrianglesGoals ■ Find the lengths of the sides of 30°–60°–90°

and 45°–45°–90° triangles.

Applications Architecture, Road Planning, Plumbing

In this 30°–60°–90° triangle,suppose the measure of theside opposite the 30° angle iss. Then the measure of thehypotenuse is 2s. You can usethe Pythagorean Theorem tofind the measure of thelonger leg of the triangle interms of s.

Let y represent the measure of theside opposite the 60° angle.

s 2 � y 2 � (2s)2

s 2 � y 2 � 4s 2

y 2 � 3s 2

y � �3s 2�

y � s�3�

2s 2s

s s60�

30� 30�

60�

In a 30°–60°–90° triangle, the measure of the hypotenuseis two times that of the leg opposite the 30° angle. The measure of the other leg is �3� times that of the leg opposite the 30° angle.

30°–60°–90°TriangleTheorem


SolutionIn a 30°–60°–90° triangle, if the leg opposite the 30° angle is s, then the other leg iss�3�, and the hypotenuse is 2s.

So, in this triangle, the leg opposite the 30° angle is 4 cm, the other leg is 4�3� cm,and the hypotenuse is 8 cm.

E x a m p l e 2

In a 30°–60°–90° triangle, the hypotenuse measures 7 in. Find themeasure of the other two sides to the nearest tenth.

Solution2s hypotenuse 2s � 7

s side opposite 30° angle s � 3.5

s�3� side opposite 60° angle s�3� � (3.5) (�3�) � 6.1

So, the missing measures of this triangle are 3.5 in. and about 6.1 in.

When you are given the measure of the leg opposite the 60° angle, you may needto rationalize the denominator.

E x a m p l e 3

In a 30°–60°–90° triangle, the measure of the leg opposite the 60° angle is5 cm. Find the measures of the other two sides in simplest radical form.

SolutionYou are given that s�3� � 5.

s � Rationalize the denominator. s � �

The measure of the hypotenuse is 2s.

Substitute the value you found for s. 2s � 2� � �

So, the missing measures are �5�3

3�� cm and �10

3�3�� cm.

E x a m p l e 4

Find the unknown measures. a. b.

SolutionIf s is the length of a side of a45°–45°–90° triangle, then the hypotenuse is s�2�.

s cm

s cm12 cm

3 in.

3 in.

10�3��

35�3��

3

5�3��

35�3�

��3� � �3�

5��3�

Lesson 10-3 Special Right Triangles 437

60�

5 cm

In a 45°–45°–90° triangle, the measure of the hypotenuseis �2� times the measure of a leg of the triangle.

45°–45°–90°TriangleTheorem

Problem SolvingTip

When you are giveninformation about ageometric figure, it is bestto draw a quick sketch ofthe figure and label anygiven measurements. Thatway, you will be able tosee what relationships canhelp you to find themissing measure.



a. Because s � 3, the measure of the hypotenuse is 3�2� in.

b. The measure of the hypotenuse is given as 12 cm. You can use the equation s�2� � 12 to solve for s.

s � ��12

2��

�12�2��

2�2�

� � �12

2�2�� Rationalize the denominator.

s � 6�2� The unknown length is 6�2� cm.

Right angles are common in our everyday life. You can find examples of rightangles in and around any building. Many streets intersect at right angles, andtrees form right angles with the ground. Therefore, right triangles and thePythagorean Theorem have many applications.

E x a m p l e 5

ROAD PLANNING Presently, to get from the ranger station tothe park exit, you must drive 6 mi north and then 9 mi east. Thepark manager is considering having a new road constructed toprovide a straight route. What will be the length of the new roadto the nearest tenth?

SolutionThe roads form a right triangle. Use the Pythagorean Theorem to find the hypotenuse, which will be the length of the new road.

Let x � the length of the new road.

62 � 92 � x 2

36 � 81 � x 2

117 � x 2

10.81665383 � x The new road will be about 10.8 mi long.

TRY THESE EXERCISES

Find the unknown side measures. First find each in simplest radical form, andthen find each to the nearest tenth.

1. 2. 3. 4.

5. A satellite dish 5 ft high is attached by a cable from its top to a point on theground 3 ft from its base. How long is the cable to the nearest tenth?

6. INVENTIONS Nefi has designed a small remote-controlled robot. He plansto test the robot’s ability to move quickly on a rectangular piece of asphaltthat measures 20 ft by 8 ft. What is the greatest distance to the nearest tenthof a foot that the robot can travel without turning?

s mm

s mm

9 mm6 yd

6 yd

10 cm60�4 in.

60�


6 mi

9 mi

proposed road

4�3� in., 8 in.;6.9 in., 8.0 in.

5�3� cm, 5 cm;8.7 cm, 5.0 cm

6�2� yd; 8.5 yd 4.5�2� mm; 6.4 mm

5.8 ft

21.5 ft



PRACTICE EXERCISES

Find the unknown measures. First find each in simplest radical form, and thenfind each to the nearest tenth.

7. 8. 9. 10.

Solve. Round answers to the nearest tenth.

11. The diagonal of a square measures 15 cm. Find the length of a side of thesquare.

12. Find the measure of the altitude of an equilateral triangle with a side thatmeasures 8 in.

13. Find the area of an equilateral triangle with a side that measures 10 cm.

14. PLUMBING A new pipe will be connected to two parallel pipes using 45° elbows. How long must the pipe be if the two parallel pipes are 2 ft apart?

15. CONSTRUCTION A beam, 10 m in length, is propped up against a building. The beam has slipped so that its base is 3 m from the wall. How far up the building does the beam reach now?


16. Use algebra to show that, if s is the measure of a leg in a 45°–45°–90°triangle, then the hypotenuse measures s�2�.

17. WRITING MATH Is this statement always, sometimes, or never true? Explainyour choice.

If one acute angle of a right triangle is half the measure of the other acute angle,then the side opposite that angle measures half the length of the hypotenuse.

18. CHAPTER INVESTIGATION Research architectural styles that wouldcomplement your school grounds. Make a list of design elements that youcould choose from in creating a new look for your school.


A card is drawn at random from a standard deck of cards. Find eachprobability. (Lesson 9-3)

19. The card is a 3, a 6, or a 9. 20. The card is a 5 or a face card.

21. The card is red and a 10. 22. The card is black and a face card.

23. The card is a club and a 7 or a jack. 24. The card is neither a spade nor a heart.

Solve each inequality and graph the solution set on a number line. (Lesson 2-6)

25. 2x � 1 � �3 26. 2(x � 1) � 6 27. �2(x � 3) �2 � 4 28. �12

�(4x � 8) � 2

5 yd

5 yd

1 m

30�8 cm

60�

10 cm

10 cm

Lesson 10-3 Special Right Triangles 439

2 ft

45�

10 m 10 m

3 m

10�2� cm; 14.1 cm 4 cm, 4�3� cm;4 cm, 6.9 cm

�3� m, 2 m;1.7 m, 2 m

5�2� yd; 7.1 yd

10.6 cm

4�3� in. or about 6.9 in.

25�3� cm2, or about 43.3 cm2

2.8 ft

9.5 m



�133� �

143�

�236�

�12

�

�216�

�216�

x � �2 x � 4 x � �6 x � 3

For 25–28, see additional answers for graphs.

Check students’ drawings.




Use a compass, straightedge and scissors to make these constructions:

1. Draw a circle, mark its center, and cut the circle out. Fold the circle in half andthen in quarters. Draw line segments along the two creases to outline one-fourth of the circle. What kind of angle is formed at the center of the circle bythe two line segments?

2. Draw another circle and cut it out. Fold the circle in half and draw along thecrease to draw a diameter. Draw a line segment to connect one endpoint ofthe diameter with any point on the circle. Then draw another line segment toconnect that point with the other endpoint of the diameter. What do younotice about the angle formed?

BUILD UNDERSTANDING

In a circle, a central angle is an angle that has its vertex atthe center of the circle. The rays of the angle are said tointercept an arc. �ABC intercepts minor arc _AC as well as major arc -AXC. A minor arc is smaller than a semicircle, anda major arc is larger than a semicircle.

The degree measure of an arc is the same as the number of degrees of thecorresponding central angle. Because m�ABC � 50, m _AC � 50 and m -AXC � 360 � 50, or 310.

E x a m p l e 1

LANDSCAPE ARCHITECTURE A portion of thecircumference of a circular pond will be tiled withhandmade tiles donated by a local school. The portion tobe tiled is represented on the plans as arc GH. Find m -GH.

Solution�GOH is the central angle that intercepts -GH. This angle measures 75°.Therefore, m -GH � 75°.

A basic assumption about arcs is the arc addition postulate.


10-4 Circles, Angles,and ArcsGoals ■ Find measures of central and inscribed angles.

■ Find measures of angles formed by intersectingsecants and tangents.

Applications Landscape Architecture, Navigation, Surveying

AC

B

X

50�

G

HO 75�

If C is a point on an arc with endpoints A and B, then

m _AC � m _BC � m -ACB.

Arc AdditionPostulate

A C

B

right angle

They form a right angle.

Animationmathmatters3.com


E x a m p l e 2

Find the measure of each arc. a. -PQR b. _SP

Solutiona. m -PQR � m _PQ � m _QR

� 45 � 40 � 85 m _PQ � 45, because m�POQ � 45

b. m _SP � 360 � (m _SR � m _RQ � m _QP)

� 360 � (120 � 40 � 45) � 155

Therefore, m -PQR � 85 and m _SP � 155.

In circle P, �ABC is an inscribed angle. It has its vertex, B, on the circle,intersects the circle at two other points, and intercepts _AC .

E x a m p l e 3

Find the measure of �QRS.

Solution�QRS intercepts _QS. m _QS � 90.

m�QRS � �12

� (90) � 45

GEOMETRY SOFTWARE Use geometrysoftware to explore the relationship betweenthe measures of an inscribed angle and itsintercepted arc.

1. Draw a circle and inscribed angle CBD.

2. Find m _CD and m�CBD.

3. Select and move point D around the circle,observing the changes to the measures of CD_ and �CBD.

In circle O, A�B�is a line segment with both endpoints on the circle; it is called a chord. CD�� is a line that intersects the circle in two places; it is called a secant. CF�� is also a secant. FG�� intersects the circle in only one point; it is called a tangent.

The angles formed by secants and tangents are also related to the degree measures of their intercepted arcs.

Lesson 10-4 Circles, Angles, and Arcs 441

C

D

B

A A

C D

B

mCD on �A � 50°m�CBD � 25°

mCD on �A � 100°m�CBD � 50°

A

B

O

C

D

E

F

G

The measure of an inscribed angle in a circle is one-half the measure of its intercepted arc.Theorem

If two secants intersect inside a circle, then the measure ofeach angle formed is equal to one-half the sum of the measuresof the arcs intercepted by the angle and its vertical angle.

Theorem

S

R

QP

O

45�

120�

40�

BC

A

P

R

Q

S

90�





E x a m p l e 4

Secants AB�� and CD�� intersect at point E inside a circle. Find themeasure of �AEC.

SolutionFor �AEC, the two arcs intercepted by the secants are _AC and _BD.

m�AEC � �12

�(m _AC � m _BD)

� �12

�(50 � 60)

� �12

�(110) � 55 So, �AEC measures 55°.

E x a m p l e 5

Find m �XZW.

Solution�XZW is formed by secant XZ�� and tangent WZ��.

m�XZW � �12

�(m _XW � m _YW)

� �12

�(130 � 40)

� �12

�(90)

� 45

So, m�XZW is 45°.

TRY THESE EXERCISES

Find x.

1. 2. 3.

168�

70�x�

110�30� x�

140�

x�O


If two secants, two tangents, or a tangent and a secantintersect outside a circle, then the measure of the angleformed is equal to one-half the positive difference of themeasures of the intercepted arcs.

Theorem

X

W

Y

Z40�

130�

60�E

A D

BC

F

G

50�

140 40 49

PRACTICE EXERCISES

Find x.

4. 5. 6.

7. NAVIGATION Explorers are searching for the site of an ancientshipwreck. From information found in a recently discovered diary,they draw a circle on a map with its center over a small island. Theydraw an inscribed angle with an intercepted arc of 80°. What is themeasure of the inscribed angle?

8. GEOMETRY SOFTWARE �ABC is anisosceles triangle inscribed in circle O.Draw the triangle so that m�BAC � 35°.What is the measure of -ADC ?

9. SURVEYING On a surveyor’s map, centralangle COD intercepts minor arc CD whichhas a measure of 100°. What is the measure of inscribed angle CRD if it also intercepts _CD?

10. WRITING MATH Write a paragraph that proves this statement: Iftwo inscribed angles intercept the same arc, the two angles arecongruent.


11. State the theorem that would apply to the measure of the angles formed bytwo chords intersecting inside a circle.

12. Describe the locus of all points x such that �XAB is a right triangle withhypotenuse A�B�.


A bag contains 5 white marbles, 8 red marbles, 7 green marbles, and 5 pinkmarbles. One marble at a time is taken from the bag and not replaced. Findeach probability. (Lesson 9-4)

13. P(pink, then white) 14. P(red, then green)

15. P(white, then red, then pink) 16. P(red, then pink, then green)

Solve each equation. (Lesson 2-5)

17. 3(4a � 6) � �12 18. �7b � 6 � �b � 12 19. �3(h � 4) � 2(3h � 3)

20. 5p � 3(p � 4) � �2 21. �2c � 6 � 2(5c � 3) 22. �12

�m � 3(m � 1) � �10

23. 12z � 3 � 6z � 9 � 3z 24. 0.5x � 4.2 � 0.2(x � 3) 25. 4(w � 3) � 2 � 5w � 8

105�

45�

x°

80�

20�

85�x�x�90�

Lesson 10-4 Circles, Angles, and Arcs 443

A

B

C

O

D

4550

40°

220°

50°


a circle with diameter A�B�

If two secants (or chords) intersect inside a circle, then themeasure of each angle formed is equal to one-half the sum of the measures of the intercepted arcs.

30

�214�

�619�

�775�

�3

745�

��23

�

�2

�2�16

1

3�12

�

�54

�

�4




PRACTICE LESSON 10-3Find the unknown side measures. First find each in simplest radical form andthen find each to the nearest tenth.

1. 2. 3. 4.

For each 30°–60°–90° triangle, find the measures of the other two sides insimplest radical form.

5. side opposite 30° angle measures 11 cm 6. hypotenuse measures 48 ft

7. side opposite 60° angle measures 5�3� in. 8. side opposite 30° angle measures �3� m

9. hypotenuse measures 37 ft 10. hypotenuse measures 6�24� yd

11. side opposite 60° angle measures 30 km 12. hypotenuse measures �13

� cm

For each 45°–45°–90° triangle, find the measures of the other two sides insimplest radical form.

13. leg measures 44 in. 14. hypotenuse measures 15�2� m

15. hypotenuse measures 8 ft 16. leg measures cm

PRACTICE LESSON 10-4Find x.

17. 18. 19.

20. 21. 22.

23. True or false: Two tangents can intersect inside a circle.

24. True or false: A chord that passes through the center of a circle is called a diameter.

x°100° 36°

x°

52°

60°

x°120° 16°

x°

95°

155°x°74°

146°x°92°

3��2�

6 yd

13 yd� 3

3 m

3 m

30°

7.5 cm45°20 in.


10�2� in.,10�2� in.;14.1 in.,14.1 in.

�15

2�3�� cm, 15 cm;

13.0 cm, 15 cm 3�2� m; 4.2 m�543�; 23.3 yd

11�3� cm, 22 cm

5 in., 10 in.

18.5 ft, 18.5�3� ft

10�3� km, 20�3� km�16

� cm, ��63�

� cm6�6� yd, 18�2� yd

3 m, 2�3� m

24 ft, 24�3� ft

44 in., 44�2� in.

4�2� ft, 4�2� ft�3�

22�

� cm, 3 cm

15 m, 15 m

46 36 125

32124

52

false

true


PRACTICE LESSON 10-1–LESSON 10-4Write each in simplest radical form. (Lesson 10-1)

25. ��38

�� 26. 27. (4�22�)(3�33�)(9�8�) 28. �(4)(12�)(18)�

Use the Pythagorean Theorem to find the unknown length. Round to the nearest tenth.(Lesson 10-2)

29. 30. 31.

Mid-Chapter QuizUse your calculator to find the value to the nearest hundredth. (Lesson 10-1)

1. �44� 2. �87�

Write each in simplest radical form. (Lesson 10-1)

3. �270� 4. (4�6�)(�5�3�) 5.

Use the Pythagorean Theorem to find the unknown length. Round your answerto the nearest tenth. (Lesson 10-2)

6. legs: x, 15 in. 7. legs: 7 cm, 9 cmhypotenuse: 17 in. hypotenuse: x

Find the missing side lengths for 30°–60°–90° and 45°–45°–90° right triangles.Round your answers to the nearest tenth. (Lesson 10-3)

Leg opposite Leg opposite30° angle 60° angle Hypotenuse

8. 4 yd

9. 6 m

Leg opposite 45° angle Hypotenuse

10. 9 ft

11. 10 in.

Find the measure of each arc or angle. (Lesson 10-4)

12. arc FG 13. arc FHG

14. angle FHG 15. angle FIG

16. angle LKF

�56��14�

17 ft

13 ft

71 m

112 m

14 mm 14 mm

9��3�


10°

20°

K

J H

O

I

G

F

110°

ML

80°

��46�

� 3�3�4752�3� 12�6�

11.0 ft132.6 m

19.8 mm

6.63 9.33

2�60�2�3�30�

8.0 in. 11.4 cm

2 yd 3.5 yd

10.4 m 12 m

12.7 ft

7.1 in.

80° 280°

40° 50°

40°

A circle graph is a good way to compare data that are parts of a whole.Each part of the whole can be changed to a percent of the whole. Thenthe percents are used to divide a circle into sectors.

P r o b l e m

URBAN PLANNING A cityplanner is looking at the resultsof a survey of housing types.She decides to make a circlegraph of the data to present tothe next city council meeting.

Solve the ProblemStep 1: Add all of the data to find the total.

9070 � 3023 � 756 � 2267 � 15,116

Step 2: Find what percent each number is of the total. Use a calculator and round percents to the nearest whole percent.

Step 3: Find the central angles that correspond to each percent.

Since there are 360° in a circle, use the percents found in Step 2to find corresponding central angle measures. For example,60% of 360° is 216°.

Step 4: Construct the graph using a compass, straightedge, andprotractor.

Start with the smallest angle and work around to the largestangle. Draw a circle and one radius. Place the protractor so thatthe 18° mark aligns with the radius. Make a mark at 18° anddraw a radius at that point.


10-5 Problem Solving Skills: Circle Graphs

Housing type Number

Single-family home 9070Two-family home 3023Three-to-six family home 756Seven or more units buildings 2267

Housing type Number Percent of total

Single-family home 9070 60%Two-family home 3023 20%Three-to-six family home 756 5%Seven or more units buildings 2267 15%

Housing type Percent of total Central angle

Single-family home 60% 216�

Two-family home 20% 72�

Three-to-six family home 5% 18�

Seven or more units buildings 15% 54�

Problem SolvingStrategies

Guess and check

Look for a pattern

Solve a simplerproblem

Make a table, chartor list

Use a picture,diagram or model

Act it out

Work backwards

Eliminate possibilities

Use an equation orformula

✔

Lesson 10-5 Problem Solving Skills: Circle Graphs 447

Place the protractor along the new radius. Make a mark at 54° and draw a radius at that point. Continue in this way around the circle.

Label each sector and write a title for the graph.

TRY THESE EXERCISES

Make a circle graph for each set of data.

1. Rodriguez Family 2. Grace SchoolMonthly Budget Sports BudgetMortgage, $750 Football, $12,000Food, $500 Baseball, $9500Car payment, $125 Soccer, $2500Utilities, $150 Swimming, $5000Credit card payment, $250 Basketball, $6000Transportation, $175Savings, $100Miscellaneous, $150

PRACTICE EXERCISES

Make a circle graph for each set of data.

3. Town Population by Age 4. Window Types Sold,Under 5 3443 Fred’s Building Supply5–13 4587 Single width, 52014–18 2428 Double width, 24119–25 4046 Bay, 18326–39 7263 Round, 2740–64 1049 Semicircular, 89Over 64 3125 Basement, 351

5. Use the data at the right to make a circle graph for family size.


Solve for x and y. (Lesson 8-5)

6. [x 3y ] � [y � 2 2x � 1] 7. [2x y ] � [y � 1 3x � 7]

8. � � � � � 9. � � � � � 10. � � � � ��2y � 16x � 2

3x2y

y � 23x � 1

x2y

3yx � 5

x � 3y � 4

3–6 family home

7 or moreunits

2-familyhome

Single familyhome

Ourtown—Housing Types

18�

54�

18�

Five-stepPlan

1 Read2 Plan3 Solve4 Answer5 Check


(5, 3)

(3, 2) (3, 5) ��13

�, 0�

(�6, �11)

For 3–5, see additional answers.

PRACTICE EXERCISES

Number of families 66,090Two persons 27,606Three persons 15,353Four persons 14,026Five persons 5938Six persons 1997Seven or more persons 1170Total persons 209,515Average per family 3.17

Family Size*

*Numbers in thousands except foraverages

Work with a partner. You will need geometry software or acompass, straightedge, and protractor.

1. Draw a circle. Draw any two chords that intersect inside thecircle and label them as shown at the right.

2. Draw A⎯

D⎯

and E⎯

C⎯

to form two triangles. Compare the angles ofthe triangles. Because all three pairs of angles are congruent,the triangles are similar.

3. Using the measures of the segments, write this proportion: �BA

DB� � �

BB

CE�.

4. Use the cross-products rule. Discuss with your partner what conclusion you canmake about the products of the lengths of segments of intersecting chords.

BUILD UNDERSTANDING

As you can see from the above activity, any two intersecting chords determinetwo similar triangles. This leads to the following theorem.

This theorem can be used to find a missing measure.

E x a m p l e 1

Find x.

SolutionSince there are two intersecting chords, theproducts of the lengths of the segments are equal.

3 � x � 2 � 12

3x � 24

x � 8

A similar type of relationship exists for intersecting secant segments. A secantsegment intersects a circle in two points and has one endpoint on the circle andone endpoint outside the circle.

10-6 Circles and SegmentsGoals ■ Find lengths of chord, secant and tangent segments.

Applications Architecture, Art, Surveying

A

D

E

C

B

If two chords intersect inside a circle, then the product of themeasures of the two segments of one chord is equal to theproduct of the measures of the two segments of the otherchord.

Theorem


3

2

12

x

Products of segments of intersecting chords are equal.

The following example will help you see what is meant by the length of a segmentand the length of the external part of the segment.

E x a m p l e 2

ARCHITECTURE An architect is redesigning a museum. One ofthe rooms will contain a circular platform to display the skeletonof a prehistoric mammal. Steel cables represented by the twosecant segments shown in the drawing to the right will be used to brace the skeleton. Find x.

SolutionH�F�and H�J�are secant segments. H�G�is the external part of H�F�, andH�I�is the external part of H�J�. The theorem refers to the length of theentire secant segment and its external part.

HF � HG � HJ � HI

18 � 8 � (6 � x)6 The length of H�J� is 6 � x.

144 � 36 � 6x

108 � 6x

18 � x

A tangent segment of a circle is a segment that has one endpoint on a circle and oneendpoint outside the circle, and the line containing the segment intersects the circle at exactly one point.

E x a m p l e 3

Find x.

SolutionT�U�is a tangent segment, U�W�is a secantsegment, and U�V�is its external segment.

(TU)2 � UW � UV

82 � (x � 4)4

64 � 4x � 16

48 � 4x

12 � x

Lesson 10-6 Circles and Segments 449

If a tangent segment and a secant segment have a commonendpoint outside a circle, then the square of the measure ofthe tangent segment is equal to the product of the measures ofthe secant segment and its external part.

Theorem

10

8

6

x

F

GH

I

J

TU

W

V

8

x

4

If two secant segments have a common endpoint outside acircle, then the product of the measures of one secantsegment and its external part is equal to the product of theother secant segment and its external part.

Theorem

CheckUnderstanding

What is the length of H⎯J?

24


Interactive Labmathmatters3.com



Another interesting property of circles and chords arises from a radiusperpendicular to a chord.

E x a m p l e 4

In circle O, radius O�R�is perpendicular to chord P�Q�at T.Find PT if PQ � 5 cm.

SolutionThe first step is to make a diagram and label it using the given facts.

The problem states that O�R�is perpendicular to P�Q�. Therefore, O�R�also bisectsP�Q�. If PQ � 5 cm, then PT � 2.5 cm.

TRY THESE EXERCISES

Find x.

1. 2. 3.

4. 5. 6.

PRACTICE EXERCISES

Find x.

7. 8. 9.

10. 11. 12.x 5

4

16

x

6

18

8

12

5

x

12

8

x

6 5

8x

6

x

x

4

8x

20

10

x

11 4

3

x

7

2

36 x4

810

x2

15


P Q

R

O

T

If a radius of a circle is perpendicular to a chord of the circle,then that radius bisects the chord.Theorem

3 5�13

� 3

123017

6 9.6 10

111227




13. ART An artist is planning a circular mosaic. The design for the mosaic hastwo chords that intersect inside the circle. Chord P�R�consists of twosegments 3 in. and 15 in. long. Chord S�T�consists of two segments, one ofwhich is 5 cm in length. Find the length of the remaining segment.

14. SURVEYING On a map of a city, two chords of a circle, TR and KL, intersectat point X. TX � 4 in., XR � 6 in., and KX � 3 in. Find the measure of K�L�.

Find x and y.

15. 16. 17.

18. GEOMETRY SOFTWARE The distance of a chord from the center of the circle is defined as the length of a segment from the center of the circle perpendicular to the chord. Draw a large circle. Then use a centimeter ruler to draw three different chords, all the same length, at different places in the circle. Find the distance from the center of the circle to each chord. What seems to be true?

19. For a circle, A�B�is a secant segment 8 cm long. Its external part is 3 cm. A�C�is another secant segment with an external part of 4 cm. What is its length?

20. CHAPTER INVESTIGATION Make a scale drawing of a new facade for a building on your school grounds. Display your original drawing and your new design.


21. WRITING MATH Look at the figure below. The theorem about theproducts of the lengths of secant segments with a common endpointoutside a circle can be proven using similar triangles. Which triangles are similar? Explain your answer.

22. WRITING MATH A tangent segment and a secant segmentare drawn to a circle from a point outside the circle. Thelength of the secant segment is 27 m, and the length of itsexternal part is 3 m. Is the length of the tangent segment arational number? Explain your answer.


Find the surface area of each figure. (Lesson 5-6)

23. 24. 25.

5.3 ft16 cm

4 cm

8.3 in.

5.1 in.3.8 in.

y

x 5

2

y

2

x

3

5

7

8

36

x

y

Lesson 10-6 Circles and Segments 451

CA B

D

E

x � 13;y � 6.5

x � 3,y � 4

x � 5, y � 12.5

9 cm

11 in.

They are all the same distance fromthe center.

6 cm



186.5 in.2 502.65 cm2352.99 ft2




PRACTICE LESSON 10-5Make a circle graph for each set of data.

1. Howe Family Budget 2. Car Types Sold 3. Fall Sports AthletesMortgage $860 Sport Coupe 50 Football 68Food $645 2-Door Sedan 35 Volleyball 24Utilities $322.50 4-Door Sedan 45 Soccer 36Insurance $107.50 Hatch Back 20 Lacrosse 38Other $215 Cross Country 34

4. Heights of Freshman 5. Technology Annual Budget 6. Park Attendance by Age�64 in. 24 New equipment $320,000 �5 171464–66 in. 36 Repair/Upgrade $80,000 5–12 229967–69 in. 72 Internet access $40,000 13–18 361770–72 in. 66 Salaries $180,000 19–55 18,196�72 in. 18 Research $100,000 �55 2991

PRACTICE LESSON 10-6Find x.

7. 8. 9.

10. 11. 12.

Classify each statement as true or false.

13. A radius of a circle bisects every chord of the circle.

14. If two secant segments have a common endpoint outside a circle and theirexternal parts are equal in length, then the chords formed by each secantinside the circle will be equal in length.

15. Two chords of a circle will never intersect at the center of the circle.

16. A secant and a tangent to a circle can intersect either outside or inside thecircle.

17. Chords of a circle which bisect each other are called diameters of the circle.

x

9

1740x

6

8

3

x

25

1810

x

5

4

7

x

9

6

5x

10 8


For 1–6, see additional answers.

4 7.5 11

75 �59

�425

false

true

false

false

true


PRACTICE LESSON 10-1–LESSON 10-6Determine if a triangle with the given sides is a right triangle. (Lesson 10-2)

18. 15 m, 36 m, 39 m 19. 10 ft, 22 ft, 30 ft 20. 16 cm, 30 cm, 34 cm

21. 18 in., 30 in., 24 in. 22. 4 yd, 4�2� yd, 4�3� yd 23. 21 m, �17� m, 21 m

Find the unknown side measures. First find each in simplest radical form, andthen find each to the nearest tenth. (Lesson 10-3)

24. 25.

26. 27.19.5 m 19.5 m

45°62 ft

60°

14 3 in.�41 cm

30°


yes

yes yes

no yes

no

41�3� cm, 82 cm;71.0 cm, 82 cm

7�3� in., 21 in.;12.1 in., 21 in.

31�2� ft, 31�2� ft,43.8 ft, 43.8 ft

�39

2�2�� m, 27.6 m

Workplace Knowhow

Career – Landscape Architects

mathmatters3.com/mathworks

Landscape architects design outdoor areas such as public parks, playgrounds,shopping centers and industrial parks. They use knowledge of the natural

environment to design areas that will complement the existing surroundings.

Landscape architects study soil, sunlight, vegetation and climate. They may workwith government officials and environmentalists to find ways to build newstructures and roads while preserving the naturalbeauty and wildlife in an area.

You are designing a circular fountain for a city park. Adiagram of the fountain is shown below. You have thefollowing measurements: DE � 4 ft 9 in., GE � 1 ft 4 in.and DF � 1 ft 6 in. Determine all measurements to thenearest tenth of a foot. Use 3.14 for pi.

1. A circular pedestal is at the center of the fountain.Find the circumference of the pedestal.

2. A wall is designed as a sitting area for park visitors.The outer edge of the wall will have a brass rim toreflect sunlight. Find the circumference of the outeredge of the wall.

3. The base of the pool, represented by the blue area ofthe diagram, will be tiled. Find the area of the base of the pool.

4. Find the total area of the fountain.

Outer Wall

Pool

Pedestal

D F G E

9.4 ft

29.8 ft

29.6 ft2

70.8 ft2

http://mathmatters3.com/mathworks

Construct a polygon using a compass and a straightedge.

1. Use a compass to draw any circle.

2. Keeping the compass open to the same radius, place the compass pointanywhere on the circle and draw a small arc that intersects the circle.

3. Place the compass point on the intersection you just made and drawanother arc but not through the starting point.

4. Continue in this way around the circle. You will have drawn six points.

5. Connect each point to the one next to it with a line segment. Measure the sides and angles of the polygon you have drawn. How would youdescribe the figure?

BUILD UNDERSTANDING

Several regular polygons can be constructed using a circle. For instance, you canconstruct an equilateral triangle using steps 1 through 4 above to draw six evenlyspaced points. Then use a straightedge to connect every other point to form thetriangle. The same basic construction can be adapted to construct a regulardodecagon, a 12-sided polygon.

E x a m p l e 1

Construct a regular dodecagon.

SolutionStep 1: Begin with a circle and mark off 6 equalarcs, as you did above.

Step 2: Connect each point to the center of thecircle. You now have 6 central angles that are allcongruent.

Step 3: Bisect three consecutive central angles. Extend each bisector sothat it intersects the circle on two sides. You should now have 12 equallyspaced points on the circle.

Step 4: Connect each point to the one next to it with a straight linesegment. The resulting figure is a regular dodecagon.

454

10-7 Constructionswith CirclesGoals ■ Construct regular polygons using circles.

■ Inscribe a circle in a polygon and circumscribe a circleabout a polygon.

Applications Architecture, Design, Art


a regular hexagon

Animationmathmatters3.com


A circumscribed polygon of a circle has every side of the polygontangent to the same circle. Any regular polygon can be circumscribedaround a circle.

E x a m p l e 2

Circumscribe a circle around this regularpentagon.

SolutionTo circumscribe a circle around any regular polygon, construct theperpendicular bisector of any two of its sides. The point ofintersection of the bisectors becomes the center of the circle. Theradius of the circle is the distance from the center to any vertex of the polygon.

An inscribed polygon has every vertex of the polygon on the samecircle. Any regular polygon can be inscribed in a circle.

E x a m p l e 3

Inscribe a circle in a square.

SolutionDraw a square. To locate the point that willbecome the center of the circle, find theintersection of the perpendicular bisectorsof any two sides. To find the radius of thecircle, use the distance from the center ofthe circle along a perpendicular bisector toa side of the polygon. Draw the circle.

Many properties of circles and parts ofcircles can be used to solve real-life problems.

E x a m p l e 4

ARCHITECTURE An architect is restoringan old house. She has found a part of awindow that may have been used in theattic. How can she figure out the size of theoriginal window from this fragment?

Lesson 10-7 Constructions with Circles 455

Problem SolvingTip

Use paper folding toconstruct a regularoctagon using a circle.

1. Draw a circle and cut itout.

2. Fold the circle intohalves, then fourthsand finally eighths.Open the circle.

3. There are now 8equally spaced pointson the circle. Connecteach point to the onenext to it with astraight line segment.The resulting figure is aregular octagon.

Reading Math

The word circumscribedcomes from Latin wordswhich mean “drawnaround.” The wordinscribed means “drawnin.” The Example 3 textsays that the circle hasbeen inscribed in thesquare. It is also correct tosay that the square hasbeen circumscribedaround the circle.



SolutionThe perpendicular bisector of a chord passes through the center of a circle. Tocomplete the circle, begin by drawing two chords. Construct the perpendicularbisector of each. The point where the bisectors intersect must be the center ofthe circle. Use the center of the circle and the radius to complete the circle.

TRY THESE EXERCISES

1. Construct a regular hexagon with a side measuring 4 cm.

2. Copy the regular octagon shown at the right.Circumscribe a circle around the octagon.

3. Copy the equilateral triangle shown at the right. Inscribe a circle in the triangle.

TALK ABOUT IT Discuss the following statements with a partner. Decide whether each statement is true or false. Explain your reasoning.

4. In Example 3, the circle is inscribed in the square.

5. It is always possible to inscribe a rhombus with sides of given lengths in a circle.

6. It is impossible to inscribe a non-regular octagon in a circle.

7. CONSTRUCTION Circular saws come in different sizes. A builderbought a saw at a second-hand sale. The saw blade was broken asshown in the picture at the right. Copy the drawing. Useconstructions to demonstrate how the builder can find the centerand complete the circle in order to find out which size ofreplacement blade to buy.

8. WRITING MATH An architect draws a circle and a diameter of the circle. Sheconstructs the perpendicular bisector of the diameter and extends it so thatit intersects the circle in two points. How could the architect use the drawingto construct a square?

PRACTICE EXERCISES

9. Construct a regular octagon.

10. Copy this regular hexagon. 11. Copy this regular pentagon.Circumscribe a circle around Inscribe a circle in the pentagon.the hexagon.


For 1–3, check students’ constructions.

true

false

false


Check students’ constructions.

Check students’ constructions. Check students’ constructions.


After completing the steps described, there are four congruent central angles and 4equally spaced points on the circle. Connect each point to the one next to it with a straight line segment. The

resulting figure is a square.

12. Draw any regular pentagon. Describe how you can use a circle to help youconstruct a regular decagon from the pentagon.

13. A circle with a radius of 5 in. is inscribed in a square. What is the perimeter ofthe square?

14. A regular hexagon is inscribed in a circle. What is the measure of each of thesix arcs of the circle?

15. Copy the regular hexagon in Exercise 10. Inscribe a circle around the hexagon.

16. ARCHITECTURE Ming Lee is an architect. Shesubmitted the following plan for a new house toher clients, James and Odetta Williams. TheWilliamses ask Ms. Lee to alter the plans so thatthe sun room is circular rather than square. CopyMing Lee’s plans onto graph paper, using pencil,compass, and straightedge. Inscribe a circlewithin the square that represents the sun room.Then erase the outline of the square.

17. DESIGN Dave O’Brien makes and designs tiles.He has a client who wants a small tabletop covered inhexagonal tiles. Dave decides to make some sketches ofdifferent designs from which the client can choose.Construct a regular hexagon measuring 2 in. on all sidesthat Dave can use as a model.

18. ART Linda Soares isdesigning a logo for the newcommunity center. Shewants to take the pentagonat the right and circumscribea circle around it while alsoinscribing a circle in thepentagon. Copy thepentagon and completeLinda’s design.

19. ARCHITECTURE Mike Whitehorse has a blueprint for a gazeboin the shape of a regular hexagon. He wants to change it so that itis about the same size, but has the shape of a regular dodecagon.How can he use a copy of the original blueprint to develop theother plan?


20. These regular polygons can be constructed using a compass andstraightedge: square, octagon, 16-gon, and 32-gon. If n is the number of sidesin a polygon, find an expression to represent the pattern.


Compute the variance and standard deviation for each set of data. Round thefinal answer to the nearest tenth if necessary. (Lesson 9-7)

21. 8, 9, 10, 11, 12, 13 22. 28, 32, 31, 36, 29, 35

23. 2, 4, 6, 3, 5, 4 24. 12, 15, 16, 11, 15, 13

25. 74, 73, 82, 80, 77, 79 26. 52, 60, 65, 62, 58, 61

Lesson 10-7 Constructions with Circles 457

stairs

Dining Room SunRoom

LivingRoom

Kitchen


40 in.

60°





2n, n � 2

2.9, 1.7

1.7, 1.3

10.25, 3.2

8.5, 2.9

3.2, 1.8

16.2, 4.0

Check students’ construction.




Chapter 10 ReviewVOCABULARY

Match the word from the list at the right with the description at the left.

1. longest side of a right triangle

2. angle that has its vertex on a circle and intersects the circle in 2 other points

3. segment with both endpoints on a circle

4. line that intersects a circle in two points

5. line that intersects a circle in only one point

6. number that cannot be written as a fraction, a terminatingdecimal, or a repeating decimal

7. number under the radical sign

8. angle that has its vertex at the center of a circle

9. polygon where every side is tangent to the same circle

10. arc with degree measure greater than 180°

LESSON 10-1 Irrational Numbers, p. 426� An irrational number cannot be written as a fraction,

terminating decimal, or repeating decimal. The square root of a number that is not a perfect square is always irrational.

� Many numbers can be written in simplest radical form using these theorems.

�a � b� � �a� � �b� ��ab

��

a�b�

�

Write each in simplest radical form.

11. �45� 12. ��2��8�� 13. �7�30��2�6��

14. ��130�� 15. �

��

86��

� 16. �2�5��2

LESSON 10-2 The Pythagorean Theorem, p. 430� In a right triangle, the two shorter sides are called the legs, and the longest

side is called the hypotenuse. In any right triangle, the Pythagorean Theoremis true: a 2 � b 2 � c 2, where a and b are the measures of the legs and c is themeasure of the hypotenuse.

Use the Pythagorean Theorem to find the unknown length. Round to thenearest tenth.17. 18. 19.

20. A pole that is 3 m high is connected by a guy wire from its top to a stake inthe ground 1.5 m from its base. How long is the wire?

5 in.14 in.

5 yd

15 yd9 m

12 m

a. central

b. chord

c. circumscribed

d. hypotenuse

e. inscribed

f. irrational

g. major

h. minor

i. radicand

j. secant

k. square root

l. tangent

Chapter 10 Review 459

LESSON 10-3 Special Right Triangles, p. 436

� In a 30°–60°–90° triangle, the measure of the hypotenuse is two times that ofthe leg opposite the 30° angle. The measure of the longer leg is �3� times theleg opposite the 30° angle.

� In a 45°–45°–90° triangle, the measure of the hypotenuse is �2� times themeasure of a leg of the triangle.

Find the unknown side measures. Give answers in simplest radical form.

21. 22. 23.

24. 25. 26.

LESSON 10-4 Circles, Angles, and Arcs, p. 440

� The measure of a central angle of a circle is the same as the measure of itsintercepted arc.

� The measure of an inscribed angle in a circle is one-half the measure of itsintercepted arc.

� If two secants intersect inside a circle, the measure of each angle formed isequal to one-half the sum of the measures of the intercepted arcs.

� If two secants, two tangents, or a tangent and a secant intersect outside acircle, then the measure of the angle formed is equal to one-half the differenceof the measures of the intercepted arcs.

Find x.

27. 28. 29.

30. 31. 32.

LESSON 10-5 Problem Solving Skills: Circle Graphs, p. 446

� Circle graphs can represent data about how a quantity is subdivided.

33. Every Elm High School student takes one foreign language. This year, 251take Italian, 478 take Spanish, 376 take French, and 50 take Japanese. Make acircle graph for this data.

215° x°145°50°

x°110°

160�50�

x�85�

15�x�x�

124�

30°

14 mm

45°6 ft

30°

4 cm

60�

6 cm

5 ft60�6 m

6 m

25°

x°

55°


34. The surface areas of the four oceans are given below. Make a circle graph for this data.

35. Students were asked to name their favorite flavor of ice cream. Eleven studentssaid vanilla, 15 students said chocolate, 8 students said strawberry, 5 studentssaid mint chip, and 3 said cookie dough. Make a circle graph for this data.

LESSON 10-6 Circles and Segments, p. 448

� If two chords intersect inside a circle, then the product of the measures of thetwo segments of one chord is equal to the product of the measures of the twosegments of the other chord.

� If two secant segments have a common endpoint outside a circle, then theproduct of the measures of one secant segment and its external part is equal to the product of the measures of the other secant segment and itsexternal part.

� If a tangent segment and a secant segment have a common endpoint outsidea circle, then the square of the measure of the tangent segment is equal to theproduct of the measures of the secant segment and its external part.

Find x.

36. 37. 38.

39. 40. 41.

LESSON 10-7 Constructions with Circles, p. 454

� An inscribed polygon has every vertex on the same circle.

� A circumscribed polygon has every side tangent to the same circle.

42. Draw a square. Circumscribe a circle around the square.

43. Construct an equilateral triangle. Inscribe a circle in the triangle.

CHAPTER INVESTIGATION

EXTENSION Make a presentation to your class of your design. Explain why youchose the design you did and how it will make your school more attractive.

9

x

43

9

6x

3

4 7

x

x

1

8

2x

3

32x 16

8

�

Ocean Surface AreaOcean Area (square miles)

Pacific 64,186,300Atlantic 33,420,000Indian 28,350,500Arctic 5,105,700

Chapter 10 AssessmentSimplify. Rationalize the denominator if necessary.

1. �700� 2. 3.

Find the unknown side measure in each right triangle. Round answers to thenearest tenth. (a and b are the measures of the legs; c is the measure of thehypotenuse.)

4. a � 8 in., b � 15 in. 5. a � 5 m c � 7 m

Find the missing measures. Give answers in simplest radical form.

6. 7. 8.

Find x.

9. 10. 11.

12. 13. 14.

15. 16. 17.

18. Construct a regular hexagon. Then inscribe a circle in it.

Solve.

19. A 20-ft ladder is placed against a building. The base of the ladder is 4 ft fromthe base of the building. How high up the building does the ladder reach, tothe nearest tenth of a foot?

20. A survey of 200 department store customers showed that 24 had traveledmore than 40 mi from home to the store, 52 traveled between 30 and 40 mifrom home, 35 traveled between 20 and 30 mi, and 89 traveled less than 20 mi from home. Draw a circle graph that shows this data.

x

2

54

x

2

133

x

6

2

x

6

2x6

4

3

80�x�

140�

x�10�

100�

x�70�

40�

80�x°

8

45�

360�

5

5

�75��3�

11��3�

Chapter 10 Assessment 461mathmatters3.com/chapter_assessment

http://mathmatters3.com/chapter_assessment


Standardized Test Practice5. Which statement is correct concerning the

probabilities of reaching into the jars withoutlooking and pulling out a blue marble?(Lesson 9-1)

greater for Jar 1 than Jar 2

greater for Jar 2 than Jar 1

equal for both jars

cannot be determined

6. What is the approximate length of a diagonal of a rectangle that is 18 ft long and 12 ft wide?(Lesson 10-2)

6.0 ft 13.4 ft

21.6 ft 30.0 ft

7. What is the ratio of the measure of �ACB to themeasure of �AOB? (Lesson 10-4)

1:1 2:1

1:2 cannot bedetermined

8. Which of the segments described could be asecant of a circle? (Lesson 10-6)

intersects exactly one point on a circle

has its endpoints on a circle

has one endpoint at the center of a circle

intersects exactly two points on a circleD

C

B

A

DC

BA

DC

BA

D

C

B

A

100marbles

Jar 1

200marbles

200marbles

50 blue50 blue20 blue20 blue

Jar 2

Test-Taking TipQuestion 6Be sure that you know and understand the PythagoreanTheorem. References to right angles, the diagonal of arectangle, or the hypotenuse of a triangle indicate that youmay need to use the Pythagorean Theorem to find the answerto the question.

Part 1 Multiple Choice

Record your answers on the answer sheetprovided by your teacher or on a sheetof paper.

1. Write all the subsets of {r, a, t, e}.(Lesson 1-1)

{r, a}, {a, t}, {r, t}, {r, e}, {a, e}, {t, e}

{r, a, t, e}, {r, a, t}, {a, t, e}, {r, a, e}, {r, t, e}

{r}, {a}, {t}, {e}, �

all of these

2. Which of the following results in a negativenumber? (Lesson 1-8)

(�2)5

5�3

�5 � (�2)5

(�3)�2 � 5

3. Which graph shows a line with a slope of �2?(Lesson 6-1)

4. What is the location of the image of A(�5, �2)if the point is translated three units to the right,translated four units up, and reflected over they-axis? (Lesson 8-4)

(�2, 2)

(2, �2)

(1, 0)

(2, 2)D

C

B

A

y

x

Dy

x

C

y

x

By

x

A

D

C

B

A

D

C

B

A

C

B

A

O

Chapter 10 Standardized Test Practice 463mathmatters3.com/standardized_test

Preparing for Standardized TestsFor test-taking strategies and morepractice, see pages 709–724.

Part 2 Short Response/Grid In

Record your answers on the answer sheetprovided by your teacher or on a sheet of paper.

9. In the figure below, AD � 25 and AB � BC.Find BC. (Lesson 3-1)

10. The angles of ascalene triangle havethe measures shownon the figure at theright. What is thevalue of x?(Lesson 4-1)

11. In the figure, what ism�PQR? (Lesson 4-3)

12. Write the ratio 18 ft to 9 yd in lowest terms.(Lesson 5-1)

13. If y � x � 2 and 2y � 3x � 19, what is thevalue of 5y? (Lesson 6-5)

14. The triangles below are similar. Find the valueof x. (Lesson 7-2)

15. On a blueprint, 1 in. represents 10 ft. Find the

actual length of a room that is 2�14

� in. long on

the blueprint. (Lesson 7-3)

16. If A � � � and B � � �,find A � 2B. (Lesson 8-5)

01

�54

02

�12

20

8�9

610 x

112

A B D

3x � 2 2x � 3

C

(2x � 10)� 40�

(4x � 20)�

Q

P R S110°

17. A drawer has 4 black socks, 16 white socks,and 2 blue socks. What is the probability ofreaching in the drawer without looking andtaking out two white socks? (Lesson 9-4)

18. How many ways can 4 students be selectedfrom a group of 7 students? (Lesson 9-5)

19. Simplify ��6�

��

3��8�� . (Lesson 10-1)

20. A 15-ft ladder is propped against a shed. If the top of the ladder rests against the shed 12 ft above ground, how far away in feet from the shed is the base of the ladder?(Lesson 10-2)

Part 3 Extended Response

Record your answers on a sheet of paper.Show your work.

21. Haley hikes 3 mi north, 7 mi east, and then 6 mi north again. (Lesson 10-2)a. Draw a diagram showing the direction and

distance of each segment of Haley’s hike.Label Haley’s starting point, her endingpoint, and the distance, in miles, of eachsegment of her hike.

b. To the nearest tenth of a mile, how far (in a straight line) is Haley from herstarting point?

c. How did your diagram help you to findHaley’s distance from her starting point?

22. The following table shows how Jack uses histime on a typical Saturday. Make a circlegraph of the data. (Lesson 10-5)

Saturday Time Use

Activity HoursJogging 1Reading 2Sleeping 9Eating 2Talking on the phone 1Time with friends 4Studying 5

http://mathmatters3.com/standardized_test

chapter 10: right triangles and circles

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