cumulative review -...

Geometry Chapter 1 Cumulative Review 33

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Cumulative ReviewChapter 1

For Exercises 1–13, choose the correct letter.

1. Find a pattern for the sequence. Use the pattern to show the next term.

1, 3, 9, 27,c

A. 81 B. 45 C. 41 D. 36

2. If XY = 12, what is the measure of XZ?

A. 35 B. 26 C. 24 D. 14

3. If EG = 42, find the value of y.

A. 5 B. 5 C. 6 D. 7

4. Find a pattern for the sequence. Use the pattern to show the next term.

S, M, T, W,c

A. S B. H C. M D. T

Use the figure at the right for Exercises 5–8.

5. What is the intersection of plane GHIJ and plane CDIH?

A. B. point H C. D.

6. Which four points are coplanar?

A. A, B, E, I B. B, C, D, E C. A, C, F, H D. E, F, I, K

7. What is another way to name plane ABEF?

A. point A B. C. plane ACDF D. plane CDIH

8. What is the intersection of and plane ACHG?

A. B. point E C. D. point B

9. Find the value of x.

A. 3 B. 4 C. 5 D. 10

10. is the 9 of &LON.

A. perpendicular bisector B. midpoint

C. segment bisector D. angle bisector

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Cumulative Review (continued)

Chapter 1

11. Which figures in the third group are widgets?

I.

II.

III.

IV.

A. I and III B. I and IV C. I, III, and IV D. I, II, and IV


A. 35 B. 45 C. 90 D. 135

13. Which property of equality or congruence justifies the following?If � , then � .

A. Reflexive Property B. Symmetric Property

C. Transitive Property D. Distributive Property

14. Open-ended Write two different patterns whose first three terms are 1, 2, 3,c Describe each pattern.

Use the figure at the right for Exercises 15–17.

15. Name a pair of parallel planes.

16. Name a pair of opposite rays with point Y as the endpoint.

17. Name all the segments shown that are parallel to .

Complete with always, sometimes, or never to make each statement true.

18. The intersection of two distinct planes is 9 one line.

19. The sum of the measures of two complementary angles is 9 180.

20. Writing Explain whether the following definition of ruler is acceptable.If it is not, write a good definition.

A ruler is a tool used for measuring.

21. Construct the perpendicular bisector of .

22. Use a protractor to draw a 45° angle. Then construct a congruent angle.

Use the graph for Exercises 23–25.

23. Find the coordinates of point C.

24. Find the coordinates of the midpoint of .

25. Find the length of .AB

CD

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Cumulative Review Geometry Chapter 134

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Cumulative ReviewChapters 1–7


1. At the local pizzeria, a small pizza has an 8-in. diameter. How much morepizza do you get if you buy a large pizza with a diameter of 16 in.?

A. 192p in.2 B. 48p in.2 C. 16p in.2 D. 8p in.2


A. 23 cm B. 23 cm

C. 26 cm D. 46 cm

3. Find the area of �ABC.

A. 10 ft2 B. 6 ft2 C. 5 ft2 D. 3 ft2

4. A right triangle has a leg that is 5 in. long and a hypotenuse that is 13 in. long. Find the length of the third side.

A. 6 in. B. 8 in. C. 10 in. D. 12 in.

5. Find the value of y.

A. - B. C. 3 D. 5

6. Find the length of segment AB with coordinates A(-4, 6) and B(-1, 2).

A. 8 B. 2.6 C. 11 D. 5

7. What can you conclude from the diagram?

A. �TSU � �TUS

B. �TSU and �TUS are supplementary.

C. m�TUV - m�STU = �TSU

D. m�TUV - m�TUS = �STU

8. At the mall, Juanita spends more than $25 to buy two of the same item.One of the items must cost at least what amount?

A. $15 B. $13 C. $12.51 D. $12.50

9. Find the length of .

A. yd B. 40p yd C. 4 yd D. yd

10. A stone pathway forms the diagonal of a square garden. A side of thegarden is 40 ft. How long is the pathway?

A. 40 ft B. 40 ft C. 46 ft D. 80 ft"3"2

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Chapters 1–7

11. Find the area of a square with radius 5 cm.

A. 25 cm2 B. 50 cm2 C. m2 D. 100 cm2

12. Find m�1.

A. 38 B. 62 C. 68 D. 71

13. Which polygons have an area of 25 cm2?

III. a square with diagonal 5 cm

III. an isosceles triangle with base 5 cm and height 10 cm

III. a rectangle with length 10 cm and perimeter 30 cm

A. I only B. II and III C. I and II D. I and III

14. Find the total area of the shaded regions.

A. (32 - 16p) cm2 B. (16p - 32) cm2

C. (64 - 32p) cm2 D. (32p - 64) cm2

15. If LN = 42, find LM.

A. 24 B. 18 C. 8 D. 6

16. Find AC.

A. 4 B. 8 C. 14 D. 28

17. Write a counterexample for the statement “If you are in a tree, thenyou used a ladder.”

18. Writing Use indirect reasoning to show that an equilateral trianglecannot have an obtuse angle.

19. Open-Ended Sketch and label two figures that have equal perimeters.

20. A regular hexagon with radius 4 cm is inscribed in a circle. Find thearea of the region between the sides of the hexagon and the circle.Answer in simplest radical form in terms of p.

21. Find the area of a regular pentagon that measures 6 m on a side and hasan apothem 4.1 m long.

22. Who is known as the father of plane geometry?

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1. Find XW.

A. 4 B. C. D.


A. �Y � �Z B. �XYW � �ZYX

C. �XWY � �XWZ D. none of the above


A. 18 B. 3.6 C. 44 D. 36.4

4. Points T and P lie on circle C. CT = 10 and m�PCT = 90. Find thelength of .

A. 10 in. B. 5p in. C. 10p in. D. 20p in.

5. Lines l1 and l2 intersect to form congruent supplementary angles. Whatcan you conclude?

A. Lines l1 and l2 are skew. B. Lines l1 and l2 are parallel.

C. Lines l1 and l2 are perpendicular. D. none of the above

6. RSTU is a rectangle. Find RP.

A. 13 B. 8.5 C. 6.5

D. cannot be determined from the information given

7. Find m�B.

A. 50 B. 80 C. 71 D. 65


A. m�P � m�N B. m�N � 65

C. m�P � 65 D. NK � NP

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Chapters 1–8

9. Refer to the triangle at the right to find the value of x.

A. 8 B. C. D. 7

10. Refer to the triangle at the right to find the value of y.

A. B. 12 C. 15 D.

11. Refer to the figure at the right to find the value of a.

A. 7 B. 21 C. 84 D. 97

12. Which quadrilateral must have perpendicular diagonals?

A. parallelogram B. rectangle C. rhombus D. isosceles trapezoid

13. Refer to the triangle at the right to find the value of x.

A. 60 B. 30 C. 30 D. 20

14. The diagonals of which quadrilateral bisect each other?

A. parallelogram B. rectangle C. rhombus D. all of the above

15. �KMP is isosceles with KM = KP. and are angle bisectors.What can you conclude?

A. �WMP is isosceles. B. and are medians.

C. and are altitudes. D. none of the above

16. Find the area of a regular decagon with a side of 6 m and a radius of 5 m.

17. Refer to the triangle at the right to find the measure of QR.

18. Writing Explain the statement “Congruent triangles are similar, butsimilar triangles need not be congruent.”

19. Open-Ended Draw a quadrilateral with congruent perpendiculardiagonals that is not a parallelogram.

20. State the contrapositive of the statement “If pigs can fly, then cows havesix legs.”

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Project Geometry Chapter 820

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Chapter 8 Project: Fractals Forever

Beginning the Chapter Project

Nature is full of shapes that are not straight lines, smooth curves, or flatsurfaces. Just look at a cloud bank or the bark on a tree. Many shapescontain patterns that repeat themselves on different scales, such as a head of cauliflower, a fern frond, and details of a coastline. Fractal geometry is the study of these irregular, self-similar shapes, called fractals.

In your project for this chapter, you will create fractals. You will learn how to do an iterative process—one in which steps are repeated in a regularcycle. Finally, you will investigate properties of fractals, including somesurprising facts about length and area.

Activities

Activity 1: DoingIn 1904, Swedish mathematician Helge von Koch created a fractal“snowflake.” Draw one by starting with a large equilateral triangle (Stage 0).

• Divide each side into three congruent segments.

• Draw an equilateral triangle on the middle segment of each side.

• Erase the middle segments on which you drew the smaller triangles.

You have just drawn Stage 1. Repeat the process to create Stage 2. (Divideall of the twelve sides of Stage 1 into three congruent segments.)

Activity 2: AnalyzingUse the Stage 3 Koch snowflake shown here and the earlier stages you madein the first activity.

• At each stage, is the snowflake equilateral?

• Suppose that each side of the original triangle is one unit. Complete thetable to find the perimeter of each stage.

• Use the table to predict the perimeter at Stage 4.

• Will there be a stage with a perimeter greater than 100 units?

Activity 3: DoingFractals have three important properties.

1. You can form them by repeating steps in a process called iteration.

2. You can continue until the steps become too small to draw. Even thenthe steps could continue in your mind; there is an infinite number ofiterations.

3. At each stage, a portion of the figure is a reduced copy of the entirefigure at previous stages. This property is called self-similarity. Thediagrams at right show Stages 0, 1, and 2 for a fractal tree.

• Make larger copies of these stages, and draw the next stage.

• Explain in words the iteration patterns for the fractal tree.Stage 1 Stage 2Stage 0

1 unit

Stage Number of Sides Length of a Side Perimeter

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Geometry Chapter 8 Project 21

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Chapter 8 Project (continued)

Activity 4: ThinkingWhat is the area of the Koch snowflake? At each stage youincrease the area by adding more and more equilateral triangles.Suppose that the area of Stage 0 is 1 square unit. Copy thediagrams, and explain why the area of the Koch snowflake willnever be greater than 2 square units.

Activity 5: ModelingRead about the Sierpinski triangle fractal described inExample 1 of the Fractal Feature on pages 430–431 ofyour textbook. You also can create a three-dimensionalSierpinski’s gasket.

• Create an equilateral tetrahedron (a solid withfour equilateral triangle faces), or Stage 0.

• Create four Stage 0 tetrahedrons and attach themto make a larger tetrahedron, Stage 1.

• Create four Stage 1 tetrahedrons and attach themto make an even larger tetrahedron, Stage 2.

• Continue the iteration, but be careful; the modelwill grow rapidly!

Finishing the Project

Prepare a report on fractals. Include the basic properties of fractals, and findnature photographs that illustrate these properties. Include a discussion ofthe perimeter and area of the Koch snowflake. Create a fractal of your owndesign. Write directions for one iteration of your fractal by listing the stepsused to create the first stage.

Reflect and Revise Ask a classmate to review your report. Ask the reviewer to test yourdirections for creating your fractal. Also, check that you have used geometricterms correctly and that your report is attractive as well as informative.

Extending the ProjectFind or create another fractal with the same properties as the Kochsnowflake. Include the fractal in your report.

Stage 01 squareunit of area

Stage 0

Stage 1

Stage 2

Take it to the NET

Visit PHSchool.com for information and links you might find helpful as you complete your project.

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Project Geometry Chapter 822

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Chapter Project ManagerChapter 8: Fractals Forever

Getting StartedRead about the project. As you work on it, you will need several sheets oftracing paper and a millimeter ruler. Keep all of your work for the project ina folder, along with this Project Manager.

Checklist Suggestions

❏ Activity 1: snowflake drawings ❏ Measure and draw carefully.

❏ Activity 2: Stage 3 snowflake analysis ❏ Define perimeter with an equation.

❏ Activity 3: fractal trees ❏ Use patterns to draw the next-stage fractal tree.

❏ Activity 4: area estimation ❏ Compare the diagrams.

❏ Activity 5: three-dimensional fractal ❏ Use sturdy building materials.

❏ report on fractals ❏ Check your work. Make sure that someone else could follow your directions.

Scoring Rubric

3 All drawings and models are accurate and neat. Calculations areaccurate. Reasoning and explanations are correct and clearly expressed.The report is complete and well organized.

2 Drawings and calculations are correctly chosen and used. There areminor errors in scale or computation. Reasoning and explanations areessentially correct but may contain awkward or unclear passages.

1 Some drawings are inaccurate. There are many computational errors.Explanations are incomplete or incorrect.

0 Major elements of the project are incomplete or missing.

Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.

Teacher’s Evaluation of Project

Activity 22 Geometry Hands-On Activities24

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Activity 22: Fractals

A fractal is created by performing infinitely many repetitions of a geometric process, thereby generating a self-similar structure that becomes more and more complex.

One of the earliest fractals, called a snowflake, was developed by the Swedish mathematician Helge von Koch in 1904. The fourth level of the von Koch snowflake is shown at the right.

Work with a partner, and follow these steps to create a simpler version of the von Koch snowflake.

Level 0 Begin with a line segment 3 in. long.

Level 1 Divide the segment into thirds.

Level 1 Replace the middle third with two segments, each having a lengththat is one-third of the length of the original segment.

Level 2 Divide each of the four segments into thirds.

Level 1 Replace the middle third of each segment with two segments, eachhaving a length that is one-third of the length of the segments atLevel 1.

Level n Divide each segment in the figure into thirds.

Level 1 Replace the middle third of each segment with two segments, eachhaving a length that is one-third of the length of the segments atLevel n - 1.

1. Create Level 3.

2. Create Level 4.

MATERIALS: Unlined paper, ruler

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Activity 22: Fractals (continued)

Work with a partner, and follow these steps to begin the construction of a fractal known as a Pythagorean tree.

3. Level 0 Near the center of a sheet of paper, begin with an isoscelesright triangle drawn in the position shown.

Draw a square, using each side of the triangle as one of thesides of the square.

4. Level 1 For each of the two smaller squares, locate the side oppositethe side that is a leg of the triangle.

On each of these opposite sides, draw an isosceles right trianglewith its hypotenuse congruent to the side of the square.

On each leg of each of the two triangles, draw a square whoseside is congruent to the leg of the triangle.

5. Level 2 Repeat Level 1 on each of the four smaller squares drawn in Level 1.

6. Level 3 Repeat Level 1 on each of the eight smaller squares drawn in Level 2.

Work with a partner, and follow these steps to begin the construction of a fractal known as a Sierpinski tree.

7. Level 0 Near the center of a sheet of paper, draw three congruent linesegments equally spaced about a point, as shown. Call eachsegment a branch.

8. Level 1 Find the midpoint of each Level 0 branch.

With a midpoint as a new common point and the outer half ofa Level 0 branch as a Level 1 branch, draw two more Level 1branches equally spaced about each midpoint.


With a midpoint as a new common point and the outer half ofa Level 1 branch as a Level 2 branch, draw two more Level 2branches, forming three congruent Level 2 branches, equallyspaced about each midpoint. There will be nine Level 2branches.


With a midpoint as a new common point and the outer half ofa Level 2 branch as a Level 3 branch, draw two more Level 3branches, forming three congruent Level 3 branches, equallyspaced about each midpoint.

Geometry Hands-On Activities Activity 22 25

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Enrichment 9-4Treasure Hunt

Dawn and Carlos discover directions to a hidden treasure on a desertedisland. The directions include distances and compass directions. The coursebegins at Hangman’s Cove. They dock their boat and construct a map fromthe cove to the treasure as shown below. They realize that the trip willconsume most of the day, so they want to find the most direct route back to their boat to avoid nightfall.

DIRECTIONS:

From Hangman’s Cove to Pirate’s Rock, proceed 45° north of east for 0.7 km. From Pirate’s Rock to Lookout Point, proceed 60° north of west for1.5 km. From Lookout Point to the treasure, proceed due west for 2 km.

1. Find the coordinates of Pirate’s Rock. Round to the nearest tenth.

2. Find the coordinates of Lookout Point by using vector addition. Roundto the nearest hundredth.

3. Find the coordinates of the treasure by using vector addition. Round tothe nearest hundredth.

4. Find the magnitude and direction that Dawn and Carlos must follow inorder to return to their boat. Round to the nearest tenth.

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Lookout Point

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Lesson 9-4 Enrichment Geometry Chapter 914

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Enrichment 9-2Measure and Discover!

Use the diagram at the right and the given information for Exercises 1–14.Measure all lengths to the nearest tenth.

Given: �ABC is inscribed in a circle whose center is point O and whose radius is .

1. Find the length of a in centimeters.

2. Find the length of b in centimeters.

3. Find the length of c in centimeters.

4. Find the measure of �A.

5. Find the measure of �B.

6. Find the measure of �C.

Find each value rounded to the nearest thousandth.

7. sin A 8. sin B 9. sin C

Find each value rounded to the nearest tenth.

10. 11. 12.

13. What can you conclude about , , and ?

14. Write your conclusion to Exercise 13 in words.

You have just discovered the Law of Sines!

Use the Law of Sines to complete Exercises 15–18. Round your answers to the nearest hundredth when necessary.

Lighthouses P and Q are located 16 km apart. A disabled ship S is sighted from both lighthouses. m�SPQ = 44 and m�SQP = 66.

15. What is the measure of �S?

16. Find the distance from the disabled ship to lighthouse P.

17. Find the distance from the disabled ship to lighthouse Q.

18. From which lighthouse should a rescue boat be sent?

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Lesson 9-2 Enrichment Geometry Chapter 912

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Geometry Chapter 9 Lesson 9-1 Enrichment 11

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Enrichment 9-1Angles of Intersection

In Exercises 39–44 on page 474 of your textbook, you learned how to findthe measure of the acute angle that a line forms with any horizontal line.Suppose that two lines intersect and that neither is horizontal. You can usethe slopes of the lines and the tangent ratio to find the measure of the angleof intersection between the two lines.

tan u =

u represents the measure of the smaller angle of intersection.In the graph at the right, u represents the measure of �1 or �3.

m1 and m2 are the slopes of the two lines.

Use the two given equations and graph for Exercises 1–6.

l1 : y = 3x

l2 : y = 5 - 2x

1. What is the slope of l1?

2. What is the slope of l2?

3. Use the formula above to find the measure of �1.

4. What is the measure of �2? Explain.



Use the two given equations for Exercises 7–10.

y = -2x + 3

x - 2y = 4

7. Graph the two equations.

8. Find the slope of each equation.

9. Find the two supplementary angles formed by the intersection of thetwo lines.

10. Was it necessary to use the formula to find the angles of intersection?Explain.

Find the two supplementary angles formed by the two lines. Round youranswers to the nearest tenth of a degree.

11. 3x + 2y = 5

4x - 3y = 1

m1 2 m21 1 m1m2

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1. Find the value of x to the nearest tenth.

A. 10.0 B. 7.0 C. 3.9 D. 3.6

2. Find the magnitude and direction of the vector.

A. 158.1 mi, 18° west of south B. 141.4 mi, 18° south of east

C. 158.1 mi, 72° south of east D. 141.4 mi, 72° south of east

3. By which postulate or theorem are the triangles congruent?

A. HL B. SAS C. SSS D. AAS

4. What is the exact length of x?

A. 10 B. 5

C. 10 D. 5

5. �TQR � �MNO. Find ON and TQ.

A. 3.75, 2.4 B. 2.4, 5

C. 16, 3.2 D. 3.75, 3.2

6. The hypotenuse of an isosceles right triangle is 6 ft long. What is thelength of one leg?

A. 6 B. 3 C. 12 D. 24

7. A triangle has angle measures of 2x + 8, 3x + 5, and 6x + 2. What arethe measures of the angles from smallest to largest?

A. 30, 58, 92 B. 33, 47, 100 C. 38, 50, 92 D. 38, 52, 90

8. Which is the greatest in �ABC?

A. sin A B. cos C

C. tan A D. tan C

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Chapters 1–9

9. Find the area of trapezoid ABCD.

A. 56 in.2 B. 48 in.2

C. 112 in.2 D. 540 in.2

10. A 15-ft ladder slides down a wall of a building until the ladder forms a 52° angle with the ground. How high up the wall will the ladder reachat this point?

A. 11.8 ft B. 24.4 ft C. 19 ft D. 11.7 ft

11. Find the area of the shaded region.

A. 9p ft2 B. 27p ft2

C. 4.5p ft2 D. 9p ft2

12. Which quadrilateral area is not equal to each of the others?

A. B. C. D.

13. Which quadrilateral has congruent diagonals?

A. trapezoid B. parallelogram C. rhombus D. rectangle

14. Find the value of x in the figure at the right.

15. What is the measure of each exterior angle of a regular hexagon?

16. Find the area to the nearest foot of a regular decagon with a radius of 8 ft and a perimeter of 50 ft.

17. Writing Why can trigonometric ratios not be used on the right angle ofa triangle?

18. Open-Ended In spherical geometry, point has the same meaning as in Euclidean geometry, but a “plane” is the surface of a sphere and a “line” is a great circle of a sphere. State a property of Euclideangeometry, and explain how it does not hold true in spherical geometry.

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Geometry Chapter 9 Lesson 9-5 Enrichment 15

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Enrichment 9-5Using the Area of a Triangle to Find Polygon Measurements

1. Two adjacent sides of a parallelogram have lengths of 8 cm and 9 cm,and the measure of the included angle is 30°. Find the area of theparallelogram.

2. The area of a parallelogram is 48 cm2. If it has a 60° angle and one ofthe sides is 8 cm long, find the length of one of the adjacent sides.

3. The area of a rhombus is 18 cm2, and one of the angles is 30°. What isthe length of a side?

4. Suppose that a triangle has two sides of lengths 3 cm and 4 cm and theincluded angle has a measure of u. Express the area of the triangle as afunction of u. What can the range of values be for u?

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