Lesson 7 2 problem solving ratios in similar polygons

440 Chapter 7 Similarity

Similar Polygons7-2

Objective To identify and apply similar polygons

A movie theater screen is in the shape of a rectangle 45 ft wide by 25 ft high. Which of the TV screen formats at the right do you think would show the most complete scene from a movie shown on the theater screen? Explain.

Similar fi gures have the same shape but not necessarily the same size. You can abbreviate is similar to with the symbol ,.

Essential Understanding You can use ratios and proportions to decide whether two polygons are similar and to fi nd unknown side lengths of similar fi gures.

You write a similarity statement with corresponding vertices in order, just as you write a congruence statement. When three or more ratios are equal, you can write an extended proportion. Th e proportion ABGH 5

BC HI 5

CD IJ 5

AD GJ is an extended proportion.

A scale factor is the ratio of corresponding linear measurements of two similar fi gures. Th e ratio of the lengths of corresponding sides BC and YZ , or more simply stated, the ratio of corresponding sides, is BCYZ 5

20 8 5

5 2. So the scale factor

of nABC to nXYZ is 52 or 5 : 2.

Key Concept Similar Polygons

Defi ne Two polygons are similar polygons if corresponding angles are congruent and if the lengths of corresponding sides are proportional.

Diagram ABCD , GHIJ

Symbols /A > /G /B > /H

/C > /I /D > /J

ABGH 5 BC HI 5

CD IJ 5

AD GJ

CB

A D

IH

G J

C X

Y

Z

B

A

ABC XYZ

15 20

25

6 8

10

Dynamic Activity Similar Polygons

A C T I V I T I

E S

D S

AAAAAAAA C

A CC

I E SSSSSSSS

DY NAMIC

Lesson Vocabulary

• similar fi gures • similar polygons • extended

proportion • scale factor • scale drawing • scale

L V L V

• s

LL VVV

• s

a W r c t

You learned about ratios in the last lesson. Can you use ratios to help you solve the problem?

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Problem 1

Got It?

Problem 2

Got It?

Lesson 7-2 Similar Polygons 441

Understanding Similarity

kMNP , kSRT

A What are the pairs of congruent angles?

/M > /S, /N > /R, and /P > /T B What is the extended proportion for the ratios of

corresponding sides?

MNSR 5 NP RT 5

MP ST

1. DEFG , HJKL. a. What are the pairs of congruent angles? b. What is the extended proportion for the ratios of the lengths of

corresponding sides?

Determining Similarity

Are the polygons similar? If they are, write a similarity statement and give the scale factor.

A JKLM and TUVW

Step 1 Identify pairs of congruent angles.

/J > /T, /K > /U, /L > /V, and /M > /W

Step 2 Compare the ratios of corresponding sides.

JK TU 5

12 6 5

2 1

KL UV 5

24 16 5

3 2

LMVW 5 24 14 5

12 7

JM TW 5

6 6 5

1 1

Corresponding sides are not proportional, so the polygons are not similar.

B kABC and kEFD

Step 1 Identify pairs of congruent angles.

/A > /D, /B > /E , and /C > /F

Step 2 Compare the ratios of corresponding sides.

ABDE 5 12 15 5

4 5

BC EF 5

16 20 5

4 5

AC DF 5

8 10 5

4 5

Yes; nABC , nDEF and the scale factor is 45 or 4 i 5.

2. Are the polygons similar? If they are, write a similarity statement and give the scale factor.

a. b.

R

ST

N

M P

M

L

J K T U

V

W

12

24

24 14

16

6 6 6

A B E F

DC

12 20

15 1016 8

K L

M Z Y

XW

N

10 20

1515 E

A B R S

T V U

CD

9

9 12

12 18

18

18

9

6

6

G

A

G

How can you use the similarity statement to write ratios of corresponding sides? Use the order of the sides in the similarity statement. MN

corresponds to SR ,

so MNSR is a ratio of

corresponding sides.

How do you identify corresponding sides? The included side between a pair of angles of one polygon corresponds to the included side between the corresponding pair of congruent angles of another polygon.

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Problem 4

Got It?

Problem 3

Got It?

442 Chapter 7 Similarity

Using Similar Polygons

Algebra ABCD M EFGD. What is the value of x?

4.5 7.2

5 11.25

FGBC 5 ED AD

Corresponding sides of similar polygons are proportional.

x7.5 5 6 9 Substitute.

9x 5 45 Cross Products Property

x 5 5 Divide each side by 9.

Th e value of x is 5. Th e correct answer is B.

3. Use the diagram in Problem 3. What is the value of y?

Using Similarity

Design Your class is making a rectangular poster for a rally. Th e poster’s design is 6 in. high by 10 in. wide. Th e space allowed for the poster is 4 ft high by 8 ft wide. What are the dimensions of the largest poster that will fi t in the space?

Step 1 Determine whether the height or width will fi ll the space fi rst.

Height: 4 ft 5 48 in. Width: 8 ft 5 96 in.

48 in. 4 6 in. 5 8 96 in. 4 10 in. 5 9.6

Th e design can be enlarged at most 8 times.

Step 2 Th e greatest height is 48 in., so fi nd the width.

648 5 10 x Corresponding sides of similar polygons are proportional.

6x 5 480 Cross Products Property

x 5 80 Divide each side by 6.

Th e largest poster is 48 in. by 80 in. or 4 ft by 623 ft.

4. Use the same poster design in Problem 4. What are the dimensions of the largest complete poster that will fi t in a space 3 ft high by 4 ft wide?

D

E

A B

7.5 6

5

x

y F

G C

9

ACan you rely on the diagram alone to set up the proportion? No, you need to use the similarity statement to identify corresponding sides in order to write ratios that are equal.

W

S You can’t solve the problem until you know which dimension fi lls the space fi rst.

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Problem 5

Got It?

Lesson 7-2 Similar Polygons 443

G

Why is it helpful to use a scale in different units? 1 cm i 200 m in the same units would be 1 cm i 20,000 cm. When solving the problem, 1200 is easier to work with than 120,000.

In a scale drawing, all lengths are proportional to their corresponding actual lengths. Th e scale is the ratio that compares each length in the scale drawing to the actual length. Th e lengths used in a scale can be in diff erent units. For example, a scale might be written as 1 cm to 50 km, 1 in. 5 100 mi, or 1 in. : 10 ft.

You can use proportions to fi nd the actual dimensions represented in a scale drawing.

Using a Scale Drawing

Design Th e diagram shows a scale drawing of the Golden Gate Bridge in San Francisco. Th e distance between the two towers is the main span. What is the actual length of the main span of the bridge?

Th e length of the main span in the scale drawing is 6.4 cm. Let s represent the main span of the bridge. Use the scale to set up a proportion.

1200 5 6.4

s length in drawing (cm)

actual length (m)

s 5 1280 Cross Products Property

Th e actual length of the main span of the bridge is 1280 m.

5. a. Use the scale drawing in Problem 5. What is the actual height of the towers above the roadway?

b. Reasoning Th e Space Needle in Seattle is 605 ft tall. A classmate wants to make a scale drawing of the Space Needle on an 812 in.–by-11 in. sheet of paper. He decides to use the scale 1 in. 5 50 ft. Is this a reasonable scale? Explain.

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444 Chapter 7 Similarity

Lesson Check

Practice and Problem-Solving Exercises

List the pairs of congruent angles and the extended proportion that relates the corresponding sides for the similar polygons.

9. RSTV , DEFG 10. nCAB , nWVT 11. KLMNP , HGFDC

R S

D E

F G

T V

A B W

VT C

K L

M

H G

F

D

C N

P

Determine whether the polygons are similar. If so, write a similarity statement and give the scale factor. If not, explain.

12. B

A

C

E

F D

1515

9

9

9

15

13.

A

B

D

4

4 4

4 6 6

6 6

E

F

G C

14. K

J L

R

Q

P

30

16 34 17

15

8

PracticeA See Problem 1.

See Problem 2.

Do you know HOW? JDRT M WHYX . Complete each statement.

1. /D > 9 2. RTYX 5 j

WX

3. Are the polygons similar? If they are, write a similarity statement and give the scale factor.

4. nFGH , nMNP. What is the value of x?

P N

M G

F

H 20

15 12 x

10

Do you UNDERSTAND? 5. Vocabulary What does the scale on a scale drawing

indicate?

6. Error Analysis Th e polygons at the right are similar. Which similarity statement is not correct? Explain.

A. TRUV , NPQU B. RUVT , QUNP

7. Reasoning Is similarity refl exive? Transitive? Symmetric? Justify your reasoning.

8. Th e triangles at the right are similar. What are three similarity statements for the triangles?

1624 18

24

12 R Q

L PD

H G

E 12

8 16

N

U Q

P

R

TV

B

A R

P

S

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Lesson 7-2 Similar Polygons 445

15. M

H T

A R U

LE

12

10

9

8 16.

S

18

14

15

24 T

H

E

L G

NI 17.

E J

RAB L

15

18 24

25.4 20

80 43

70

6321.3

Algebra Th e polygons are similar. Find the value of each variable.

18.

6

8

5 10

y

x

6

8

5 10

y

x 19.

12 9

6 3.5

6

y z

x 20. x

15

25.5

37

30 y

21. Web Page Design Th e space allowed for the mascot on a school’s Web page is 120 pixels wide by 90 pixels high. Its digital image is 500 pixels wide by 375 pixels high. What is the largest image of the mascot that will fi t on the Web page?

22. Art Th e design for a mural is 16 in. wide and 9 in. high. What are the dimensions of the largest possible complete mural that can be painted on a wall 24 ft wide by 14 ft high?

23. Architecture You want to make a scale drawing of New York City’s Empire State Building using the scale 1 in. 5 250 ft. If the building is 1250 ft tall, how tall should you make the building in your scale drawing?

24. Cartography A cartographer is making a map of Pennsylvania. She uses the scale 1 in. = 10 mi. Th e actual distance between Harrisburg and Philadelphia is about 95 mi. How far apart should she place the two cities on the map?

In the diagram below, kDFG M kHKM . Find each of the following.

25. the scale factor of nHKM to nDFG 26. m/K

27. GDMH 28. MK 29. GD

30. Flags A company produces a standard-size U.S. fl ag that is 3 ft by 5 ft. Th e company also produces a giant-size fl ag that is similar to the standard-size fl ag. If the shorter side of the giant-size fl ag is 36 ft, what is the length of its longer side?

31. a. Coordinate Geometry What are the measures of /A, /ABC , /BCD, /CDA, /E , /F , and /G? Explain.

b. What are the lengths of AB, BC , CD, DA, AE , EF , FG, and AG? c. Is ABCD similar to AEFG? Justify your answer.

See Problem 3.

See Problem 4.

See Problem 5.

ApplyB

30

27.5 15

18 59

70

D F H K

M

G

y

O x

6

2

1 4 6

E

F

G

C

D A

B

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446 Chapter 7 Similarity

32. Think About a Plan Th e Davis family is planning to drive from San Antonio to Houston. About how far will they have to drive?

• How can you fi nd the distance between the two cities on the map?

• What proportion can you set up to solve the problem?

33. Reasoning Two polygons have corresponding side lengths that are proportional. Can you conclude that the polygons are similar? Justify your reasoning.

34. Writing Explain why two congruent fi gures must also be similar. Include scale factor in your explanation.

35. nJLK and nRTS are similar. Th e scale factor of nJLK to nRTS is 3 i 1. What is the scale factor of nRTS to nJLK ?

36. Open-Ended Draw and label two diff erent similar quadrilaterals. Write a similarity statement for each and give the scale factor.

Algebra Find the value of x. Give the scale factor of the polygons.

37. nWLJ , nQBV 38. GKNM , VRPT

J

L

W x 6

x

B

VQ 8

5 M

G K

R

P

T

VN

3

3

4

4 3x 2

x 4

8.4

6.3

Sports Choose a scale and make a scale drawing of each rectangular playing surface.

39. A soccer fi eld is 110 yd by 60 yd. 40. A volleyball court is 60 ft by 30 ft.

41. A tennis court is 78 ft by 36 ft. 42. A football fi eld is 360 ft by 160 ft.

Determine whether each statement is always, sometimes, or never true.

43. Any two regular pentagons are similar. 44. A hexagon and a triangle are similar.

45. A square and a rhombus are similar. 46. Two similar rectangles are congruent.

47. Architecture Th e scale drawing at the right is part of a fl oor plan for a home. Th e scale is 1 cm 5 10 ft. What are the actual dimensions of the family room?

48. Th e lengths of the sides of a triangle are in the extended ratio 2 i 3 i 4. Th e perimeter of the triangle is 54 in.

a. Th e length of the shortest side of a similar triangle is 16 in. What are the lengths of the other two sides of this triangle?

b. Compare the ratio of the perimeters of the two triangles to their scale factor. What do you notice?

Master bedroom

Family room

Dining Kitchen

ChallengeC

San Antonio Galveston

Corpus Christi

Brownsville

Laredo Scale 1 cm : 112 km

Del Rio

Austin Houston

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Lesson 7-2 Similar Polygons 447

49. In rectangle BCEG, BC i CE = 2 i 3. In rectangle LJAW, LJ i JA = 2 i 3. Show that BCEG , LJAW.

50. Prove the following statement: If nABC , nDEF and nDEF , nGHK , then nABC , nGHK.

Mixed Review

If x7 5 y 9, complete each statement using the properties of proportions.

55. 9x 5 j 56. xy 5 j j

57. x 1 77 5 j j

Use the diagram for Exercises 58–61.

58. Name the isosceles triangles in the fi gure.

59. CD > 9 > 9

60. AE 5 9 61. m/A 5 9

Get Ready! To prepare for Lesson 7-3, do Exercises 62–64.

How can you prove that the triangles are congruent?

62. 63. 64.

See Lesson 7-1.

See Lesson 4-5.A

C

B

D E

F

5

42 3

See Lessons 4-2 and 4-3.

Standardized Test Prep

51. PQRS , JKLM with a scale factor of 4 i 3. QR 5 8 cm. What is the value of KL?

6 cm 8 cm 1023 cm 24 cm

52. Which of the following is NOT a property of an isosceles trapezoid?

Th e base angles are congruent. Th e diagonals are perpendicular.

Th e legs are congruent. Th e diagonals are congruent.

53. In the diagram at the right, what is m&1?

45 75 125 135

54. A high school community-action club plans to build a circular play area in a city park. Th e club members need to buy materials to enclose the area and sand to fi ll the area. For a 9-ft-diameter play area, what will be the circumference and area rounded to the nearest hundredth?

SAT/ACT

60

75 1Short Response

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