Cumulative Review

Name

Class

Date

Cumulative Review

Chapter 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?

X

2x 2 Y

3x 1

A. 35

B. 26

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

Z

C. 24

D. 14

E

3y 4

F

5y 2

G

A. 5

B.

5

1 2

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. Wh*at is) the intersection of plane GHIJ and* pl)ane CDIH?

*)

A. GH

B. point H

C. CD

D. HI

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* na)me plane ABEF?

A. point A

B. CD

C. plane ACDF

* )

8. What is the intersection of BE and plane A* CH) G?

A. BC

B. point E

C. BK

D. plane CDIH D. point B

C B A

H G

DK FE

I J

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9. Find the value of x. A. 3 B. 4 C. 5

D. 10

Q (4x 5)? (8x)?

R

)

10. OM is the 9 of &LON. A. perpendicular bisector C. segment bisector

ST

B. midpoint D. angle bisector

O N

M L

Geometry Chapter 1

Cumulative Review

33

Name

Class

Date

Cumulative Review (continued)

Chapter 1

11. Which figures in the third group are widgets?

widgets A. I and III

I.

II.

III.

not widgets

IV.

B. I and IV

C. I, III, and IV

D. I, II, and IV

12. Find the value of x.

A. 35

B. 45

C. 90

D. 135

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

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.

(3x) x

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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.

R Y

S

Q

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

T

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 EF.

y

E

F

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

4 4 A 22

V W

U X

B

Use the graph for Exercises 23?25. 23. Find the coordinates of point C. 24. Find the coordinates of the midpoint of CD. 25. Find the length of AB.

44 22 O

C

22

4 4

22 4 4 x D

34

Cumulative Review

Geometry Chapter 1

Name

Class

Date

Cumulative Review

Chapters 1?7

For Exercises 1?16, choose the correct letter.

1. At the local pizzeria, a small pizza has an 8-in. diameter. How much more

pizza 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

2. Find the value of x. A. 23 cm C. 26"3 cm

3. Find the area of ABC.

A. 10 ft2

B. 6 ft2

B. 23"2 cm D. 46"2 cm

46 cm

C. 5 ft2

x

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.

3 ft B

3 ft

A

C

4 ft

5. Find the value of y.

A. -31

B.

1 3

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

(13y 1)

7. What can you conclude from the diagram? A. TSU TUS B. TSU and TUS are supplementary. C. mTUV - mSTU = TSU D. mTUV - mTUS = STU

T

S

U

V

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

0 9. Find the length of AB .

A.

8p 5

yd

B. 40p yd

C. 4 yd

D.

4p 5

yd

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

A. 40 ft

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

A 4 yd 72 B

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Geometry Chapter 7

Cumulative Review

37

Name

Class

Date

Cumulative Review (continued)

Chapters 1?7

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

A. 25 cm2

B. 50 cm2

C. 50"2 m2

D. 100 cm2

12. Find m1. A. 38

B. 62

C. 68

D. 71

13. Which polygons have an area of 25 cm2?

III. a square with diagonal 5"2 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

1 38

14. Find the total area of the shaded regions.

A. (32 - 16p) cm2 C. (64 - 32p) cm2

B. (16p - 32) cm2

8 ft

D. (32p - 64) cm2

15. If LN = 42, find LM.

A. 24

B. 18

C. 8

D. 6

3x

2x 2

L

M

N

16. Find AC.

B

C

A. 4

B. 8

C. 14

D. 28

17. Write a counterexample for the statement "If you are in a tree, then

you used a ladder."

A

y

6 3y

4

18. Writing Use indirect reasoning to show that an equilateral triangle cannot 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 the area 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 has an apothem 4.1 m long.

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

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38

Cumulative Review

Geometry Chapter 7

Name

Class

Cumulative Review

Chapters 1?8

For Exercises 1?15, choose the correct letter.

1. Find XW.

A. 4

B.

9 5

C.

25 3

D. "15

2. What can you conclude from the diagram?

A. Y Z

B. XYW ZYX

C. XWY XWZ

D. none of the above

Date

X Y 3W 5 Z

3. Find the value of x.

A. 18

B. 3.6

C. 44

D. 36.4

(3x 10)

(2x 8)

4. Points T a0 nd P lie on circle C. CT = 10 and mPCT = 90. Find the length of TP.

A. 10"2 in.

B. 5p in.

C. 10p in.

D. 20p in.

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5. Lines l1 and l2 intersect to form congruent supplementary angles. What can 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

R

S

6. RSTU is a rectangle. Find RP.

A. 13

B. 8.5

C. 6.5

5 P

D. cannot be determined from the information given

U

12

T

A

7. Find mB. A. 50

B. 80

C. 71

D. 65

8. What can you conclude from the diagram?

A. mP mN

B. mN 65

C. mP 65

D. NK NP

P

8 50 8

B5C N

6

65

8

K

Geometry Chapter 8

Cumulative Review

31

Name

Class

Date

Cumulative Review (continued)

Chapters 1?8

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

A. 8

B.

49 8

C.

56 5

D. 7

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

A.

35 8

B. 12

C. 15

D.

75 8

78 x5 7

y

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

A. 7

B. 21

C. 84

D. 97

(4a 13) (3a 20)

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"3

C. 30"2

D. 20"3

x 30

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. MX and PY are angle bisectors.

What can you conclude?

A. WM) P is iso) sceles.

C. MX and PY are altitudes.

))

B. MX and PY are medians. 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.

60

K

Y

X

W

M

P

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

18. Writing Explain the statement "Congruent triangles are similar, but similar triangles need not be congruent."

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

M

15

24

Q

R 16 T

20. State the contrapositive of the statement "If pigs can fly, then cows have six legs."

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32

Cumulative Review

Geometry Chapter 8

Name

Class

Date

Chapter 8 Project: Fractals Forever

Beginning the Chapter Project

Nature is full of shapes that are not straight lines, smooth curves, or flat surfaces. Just look at a cloud bank or the bark on a tree. Many shapes contain 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 regular cycle. Finally, you will investigate properties of fractals, including some surprising facts about length and area.

Activities

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

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. (Divide all of the twelve sides of Stage 1 into three congruent segments.)

Stage 1

Activity 2: Analyzing

Use the Stage 3 Koch snowflake shown here and the earlier stages you made in the first activity.

? At each stage, is the snowflake equilateral?

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

Stage 3

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

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

Activity 3: Doing Fractals have three important properties.

Stage 0 1 2 3

Number of Sides 3 48

Length of a Side

1 1 3

Perimeter 3

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 then the steps could continue in your mind; there is an infinite number of iterations.

3. At each stage, a portion of the figure is a reduced copy of the entire figure at previous stages. This property is called self-similarity. The 1 unit diagrams 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 0 Stage 1 Stage 2

20

Project

Geometry Chapter 8

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Name

Class

Date

Chapter 8 Project (continued)

Activity 4: Thinking What is the area of the Koch snowflake? At each stage you increase the area by adding more and more equilateral triangles. Suppose that the area of Stage 0 is 1 square unit. Copy the diagrams, and explain why the area of the Koch snowflake will never be greater than 2 square units.

Stage 0 1 square unit of area

Activity 5: Modeling Read about the Sierpinski triangle fractal described in Example 1 of the Fractal Feature on pages 430?431 of your textbook. You also can create a three-dimensional Sierpinski's gasket.

? Create an equilateral tetrahedron (a solid with four equilateral triangle faces), or Stage 0.

? Create four Stage 0 tetrahedrons and attach them to make a larger tetrahedron, Stage 1.

? Create four Stage 1 tetrahedrons and attach them to make an even larger tetrahedron, Stage 2.

? Continue the iteration, but be careful; the model will grow rapidly!

Stage 0 Stage 1

Finishing the Project

Prepare a report on fractals. Include the basic properties of fractals, and find nature photographs that illustrate these properties. Include a discussion of the perimeter and area of the Koch snowflake. Create a fractal of your own design. Write directions for one iteration of your fractal by listing the steps used to create the first stage.

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

Extending the Project Find or create another fractal with the same properties as the Koch snowflake. Include the fractal in your report.

Stage 2

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Take it to the NET

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

Geometry Chapter 8

Project

21

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