506 Chapter 10 Geometric Figures

Groundhog Day A 16 inch tall groundhog emerges on Groundhog Day near a tree and sees its shadow. The length of the groundhog’s shadow is 5 inches, and the length of the tree’s shadow is 35 inches. What is the height of the tree?

This problem can be solved using similar triangles, as you will see in Example 2. Because the ratios of the lengths of corresponding sides are equal in similar polygons, you can write and solve proportions to find unknown lengths.

In the Real World

Word WatchReview Wordsproportion, p. 387similar polygons, p. 502

B E F O R E Now W H Y ?

Using Proportions with

Similar Polygons

You identified correspondingparts of similar polygons.

You’ll use similar triangles tofind lengths indirectly.

So you can find the height of theGateway Arch, as in Ex. 10.

with Review

Need help writing andsolving proportions? Seepp. 387 and 394.

E X A M P L E 1 Finding an Unknown Length

Quadrilaterals ABCD and EFGH are similar. Find FG.

Solution

Use the ratios of the lengths of corresponding sides to write a proportion involving the unknown length.

�AEH

D� � �

BFG

C� Write proportion involving FG.

�32

20� � �

4x0� Substitute known values.

32x � 20 p 40 Cross products property

�3322x

� � �20

3p2

40� Divide each side by 32.

x � 25 Simplify.

ANSWER The length of FG*& is 25 centimeters.

E H

GF

20 cm

x

B

A

D C

40 cm32 cm

Indirect Measurement Because the sun’s rays hit objects that areperpendicular to the ground at the same angle, similar triangles areformed by objects and their shadows. You can use these similar trianglesto find lengths that are difficult to measure directly.

Find the unknown length x given that the polygons are similar.

1. 2.

U V

T W

x

27 in. Q R

SP

35 in.

21 in.

48 m

35 m

24 m

40 m

L J

x

K

42 mF

G

H

Your turn now

E X A M P L E 2 Making an Indirect Measurement

Groundhog Day You can use indirectmeasurement to find the height of the tree described at the top of page 506.

Solution

Use the ratios of the lengths ofcorresponding sides to write a proportioninvolving the unknown height h.

�

�1h6� � �

355� Substitute known values.

16 p �1h6� � 16 p �

355� Multiply each side by 16.

h � 16 p 7 Simplify fraction.

h � 112 Multiply.

ANSWER The tree has a height of 112 inches, or 9 feet 4 inches.

Length of tree’s shadow����Length of groundhog’s shadow

Height of tree���Height of groundhog

Use indirect measurement to solve the problem.

3. The shadow cast by a lighthouse is 30 feet long. At the same time, the shadow cast by a 4 foot tall sign is 3 feet long. How tall is thelighthouse?

Your turn now

Lesson 10.6 Using Proportions with Similar Polygons 507

35 in.

5 in.

16 in.h

ExercisesMore Practice, p. 714

508 Chapter 10 Geometric Figures

Find the unknown length x given that the polygons are similar.

5. 6.

7. 8.

9. Dinosaurs A person who is 6 feet tall stands next to a life-size model ofa dinosaur. The shadow cast by the person is 4 feet long. At the sametime, the shadow cast by the dinosaur model is 12 feet long. How tall isthe dinosaur model?

x

C

D

B

A

75 m 36 m

G

H

F

E

45 m

F

E H

G

20 in.

16 in.

K

J M

L

x

24 in.

U

x

35 ft

S

28 ft

P

R

TQ

40 ft32 ft

24 ft

C E

Ax 12 cm

FB

D

16 cm9 cm

Practice and Problem Solving

1. Vocabulary Copy and complete: An equation that states that two ratiosare equivalent is called a(n) _?_.

Find the unknown length x given that the polygons are similar.

2. 3.

4. Guided Problem Solving A tourist who is 5 feet tall stands next to aSaguaro cactus. The length of the tourist’s shadow is 2 feet, and thelength of the cactus’s shadow is 13 feet. How tall is the cactus?

Draw a diagram to represent the situation.

Write a proportion involving the unknown height of the cactus.

Solve the proportion. 3

2

1

Q R

P Sx

16 in.

K L

MJ3 in.

12 in.A B

C

F

3 cm

x

9 cm

D E6 cm

9 cm

6 cm

Getting Ready to Practice

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Example Exercises1 5–8, 11–122 9–10

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Lesson 10.6 Using Proportions with Similar Polygons 509

10. Gateway Arch A boy who is 5 feet tall stands under the Gateway Archin St. Louis and casts a shadow that is 1 foot long. At the same time, theshadow of the arch is 126 feet long. How tall is the arch?

Find the unknown lengths given that the polygons are similar.

11. 12.

13. Writing Suppose you want to find the height of your school building.Describe a method for finding the height that involves indirectmeasurement.

14. Challenge Find the unknownlength given that TRST STQSP.

27 yd C

Dx

B

A15 yd y

18 yd FE

24 yd1.15 m

1.3 m C

D

x

B

A

2 m

z

y G

H

2.16 m

F

E

2.4 m

For the given angle measure, find the measure of a complementaryangle, if possible. (Lesson 10.1)

15. 39� 16. 63� 17. 75� 18. 100�

19. Given that TRSTcTCDE, name the corresponding sides andcorresponding angles. (Lesson 10.5)

Use a compass to draw a circle with the givenradius.

20. 3 in. 21. 6 cm 22. 4.5 cm 23. 2 �14

� in.

Basic Skills

Mixed Review

24. Multiple Choice Rectangles RSTU and LMNP are similar. RectangleRSTU has a length of 7 cm and a width of 4 cm. Rectangle LMNP has a length of 21 cm. What is the width of rectangle LMNP ?

A. 1.3� cm B. 3 cm C. 9 cm D. 12 cm

25. Short Response A man who is 6 feet tall stands next to a street sign.The man’s shadow is 4 feet long. At the same time, the sign’s shadow is 6 feet long. Write and solve a proportion to find the height of the sign.

Test-Taking Practice

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■ Gateway ArchEach leg of the Gateway Archis an equilateral triangle incross section. At ground level, each side of the twotriangular legs has a length of 54 feet. What is theperimeter of each leg atground level?

Tourism

6 in. QR xS

T

P

8 in.12 in.