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Determine whether the given polygons aresimilar. Explain.

1.

SOLUTION:

Yes; map DEFG to JKLM using a dilation centeredat the origin with scale factor 3 followed by atranslation along (6, 0).

ANSWER:

Yes; map DEFG to JKLM using a dilation centeredat the origin with scale factor 3 followed by atranslation along (6, 0).

2.

SOLUTION:

The angles are not the same, so the polygons do nothave the same shape, so there are no similaritytransformations between the figures.

ANSWER:

No; the angles are not the same, so the polygons donot have the same shape, so there are no similaritytransformations between the figures.

List all pairs of congruent angles, and write aproportion that relates the corresponding sidesfor each pair of similar polygons.

3.

SOLUTION:

The order of vertices in a similarity statementidentifies the corresponding angles and sides. Sincewe know that , we can take thecorresponding angles of this statement and set themcongruent to each other. Then, since thecorresponding sides of similar triangles areproportional to each other, we can write a proportionthat relates the corresponding sides to each other.

ANSWER:

A Z, B Y, C X;

eSolutions Manual - Powered by Cognero Page 1

7-2 Similar Polygons

4. JKLM TSRQ

SOLUTION:

The order of vertices in a similarity statementidentifies the corresponding angles and sides. Sincewe know that , we can take thecorresponding angles of this statement and set themcongruent to each other. Then, since thecorresponding sides of similar polygons areproportional to each other, we can write a proportionthat relates the corresponding sides to each other.

ANSWER:

J T, K S, M Q; L R;

Determine whether each pair of figures issimilar. If so, write thesimilarity statement and scale factor. If not,explain your reasoning.

5.

SOLUTION:

Step 1: Compare corresponding angles: Since all of the angles in the polygons are rightangles, they are all congruent to each other.Therefore, corresponding angles are congruent. Step 2: Compare corresponding sides:

Since , the figures are not similar.

ANSWER:

no;



6.

SOLUTION:

Step 1: Compare corresponding angles:

Therefore, . Step 2: Compare corresponding sides:

Therefore and the

corresponding sides are proportional. So since

and

The scale factor is

ANSWER:

Yes; since and

scale factor:

Each pair of polygons is similar. Find the valueof x.

7.

SOLUTION:

Use the corresponding side lengths to write aproportion.

Solve for x.

ANSWER:

6

8.

SOLUTION:


Solve for x.

ANSWER:

16



9. DESIGN On the blueprint of the apartment shown,the balcony measures 1 inch wide by 1.75 incheslong. If the actual length of the balcony is 7 feet,what is the perimeter of the balcony?

SOLUTION:

Write a proportion using the given information. Let xbe the actual width of balcony. 1 foot = 12 inches. So,7 feet = 84 inches.

So, width = 48 inches or 4 feet.

Therefore, the perimeter of the balcony is 22 ft.

ANSWER:

22 ft

Determine whether the given polygons aresimilar. Explain.

10.

SOLUTION: Yes, map triangle GHJ to triangle KLM using adilation centered at the origin with scale factor 3followed by a counterclockwise rotation of 90degrees about the origin.

ANSWER: Yes, map triangle GHJ to triangle KLM using adilation centered at the origin with scale factor 3followed by a counterclockwise rotation of 90degrees about the origin.

11.

SOLUTION: Yes; map PQRS to WXYZ using a dilation centered atthe origin with scale factor 0.5 followed by atranslation along (13, 2).

ANSWER:

Yes; map PQRS to WXYZ using a dilation centeredat the origin with scale factor 0.5 followed by atranslation along (13, 2).



List all pairs of congruent angles, and write aproportion that relates the corresponding sidesfor each pair of similar polygons.

12.

SOLUTION:

The order of vertices in a similarity statementidentifies the corresponding angles and sides. Sincewe know that , we can take thecorresponding angles of this statement and set themcongruent to each other. Then, since thecorresponding sides of similar triangles areproportional to each other, we can write a proportionthat relates the corresponding sides to each other.

ANSWER:

13.

SOLUTION:

The order of vertices in a similarity statementidentifies the corresponding angles and sides. Sincewe know that , we can take thecorresponding angles of this statement and set themcongruent to each other. Then, since thecorresponding sides of similar polygons areproportional to each other, we can write a proportionthat relates the corresponding sides to each other.

ANSWER:

J P, F S, M T; H Q,



CONSTRUCT ARGUMENTS Determinewhether each pair of figures is similar. If so,write the similarity statement and scale factor. Ifnot, explain your reasoning.

14.

SOLUTION:

Compare corresponding angles:

Since, , then the trianglesare not similar.

ANSWER:

no; L W

15.

SOLUTION:

Since the ( by SAS trianglecongruence theorem), then their corresponding partsare congruent Step 1: Compare corresponding angles:

Step 2: Compare corresponding sides:

Therefore and the

corresponding sides are proportional, with a scale

factor of . So, because The scale factor is 1.

ANSWER:

Yes; because scale factor: 1.



16. GAMES The dimensions of a hockey rink are 200feet by 85 feet. Are the hockey rink and the airhockey table shown similar? Explain your reasoning.

SOLUTION:

No, the ratio of the dimensions of the hockey rink andair hockey table are not the same.

The ratio of their lengths is about 2 and their widths isabout 1.7.

ANSWER:

No; sample answer: The ratio of the lengths of thehockey rink and air hockey table is about 2 and theratio of the widths is about 1.7.

17. COMPUTERS The dimensions of a 17-inch flat

panel computer screen are approximately by

inches. The dimensions of a 19-inch flat panel

computer screen are approximately by 12

inches. To the nearest tenth, are the computerscreens similar? Explain your reasoning.

SOLUTION:

Yes; sample answer: In order to determine if twopolygons are similar, you must compare the ratios oftheir corresponding sides.

The ratio of the longer dimensions of the screens isapproximately 1.1 and the ratio of the shorterdimensions of the screens is approximately 1.1. Sincethe table and the rink are both rectangles, we knowall of their angles are congruent to each other and,since the ratio of their corresponding sides are thesame, the shapes are similar.

ANSWER:

Yes; sample answer: The ratio of the longerdimensions of the screens is approximately 1.1 andthe ratio of the shorter dimensions of the screens isapproximately 1.1.



REGULARITY Each pair of polygons is similar.Find the value of x.

18.

SOLUTION:


Solve for x.

ANSWER:

7

19.

SOLUTION:


Solve for x.

ANSWER:

5

20.

SOLUTION:


Solve for x.

ANSWER:

2

21.

SOLUTION:


Solve for x.

ANSWER:

3



22. Rectangle ABCD has a width of 8 yards and a lengthof 20 yards. Rectangle QRST, which is similar torectangle ABCD, has a length of 40 yards. Find thescale factor of rectangle ABCD to rectangle QRSTand the perimeter of each rectangle.

SOLUTION:

Let x be the width of rectangle QRST.Use the corresponding side lengths to write aproportion.

Solve for x.

Therefore, the scale factor is 1:2.

Therefore, the perimeter of rectangle ABCD is 56yards.

Therefore, the perimeter of rectangle QRST is 112yards.

ANSWER:

1: 2; perimeter of ABCD = 56 yd, perimeter of QRST= 112 yd

Find the perimeter of the given triangle.23. AB = 5, BC = 6, AC = 7,

and DE = 3

SOLUTION:


Solve for EF.

Solve for DF.

ANSWER:

10.8



24. ST = 6, WX = 5, and theperimeter of

SOLUTION:

The scale factor of triangle SRT to triangle WZX is

or .

Use the perimeter of triangle SRT and the scalefactor to write a proportion and then substitute in thevalue of the perimeter of triangle SRT and solve forthe perimeter of triangle WZX.

ANSWER:

12.5

25. ADEG is aparallelogram, CH = 7, FH = 10, FE = 11, and EH =6

SOLUTION:


Solve for BH.

Solve for BC.

ANSWER:

18.9



26. perimeter of DF = 6, FC = 8

SOLUTION:

The scale factor of triangle CBF to triangle DEF is

or .

Use the perimeter of triangle CBF and the scalefactor to write a proportion.Then, substitute the givenvalue of the perimeter of triangle CBF and solve forthe perimeter of triangle DEF.

ANSWER:

20.25

27. Two similar rectangles have a scale factor of 2: 4.The perimeter of the large rectangle is 80 meters.Find the perimeter of the small rectangle.

SOLUTION:

Use the perimeter of the large rectangle and thescale factor to write a proportion. Then, substitute inthe given value of the large rectangle and solve forthe perimeter of the small rectangle.

ANSWER:

40 m



28. Two similar squares have a scale factor of 3: 2. Theperimeter of the small rectangle is 50 feet. Find theperimeter of the large rectangle.

SOLUTION:

Use the perimeter of the large rectangle and thescale factor to write a proportion.Then, substitute thegiven value of the perimeter of the small rectangleinto the proportion. Solve for the perimeter of thelarge rectangle.

Thus, the perimeter of the large rectangle is 75 ft.

ANSWER:

75 ft

29. PROOF Write a paragraph proof of Theorem 7.1.

Given: and

Prove:

SOLUTION:

For this proof, because we know that , we can write a similarity statement

relating the corresponding sides, such as Notice that we also know that

. Therefore, we can set each of these ratios

equal to , due to the Transitive property, and isolate

each individual side length of into separateequations. We are doing this because we want to getboth perimeter formulas in terms of the same sidelengths. Then, substitute each side relationship intothe perimeter formula for each triangle and simplify.

Given: and

Prove:

Proof: Because

So Cross products yield

and

Using substitution, the perimeter of

or

The ratio of the two perimeters

=

ANSWER:

Given: and

Prove:



Proof: Because

So Cross products yield

and

Using substitution, the perimeter of

or

The ratio of the two perimeters

=

30. PHOTOS You are enlarging the photo shown foryour school yearbook. If the dimensions of the

original photo are inches by inches and the

scale factor of the old photo to the new photo is 2: 3,what are the dimensions of the new photo?Refer to Page 509.

SOLUTION:

Let the unknown width be w and the unknown lengthbe . Since the scale factor of the original photo to the newphoto is 2:3, we can write proportions comparing theirdimensions. Form two proportions with the given information andsolve for w and .

Therefore, the dimensions of the new photo are

inches by inches.

ANSWER:

in. by in.



31. CHANGING DIMENSIONS Rectangle QRST issimilar to rectangle JKLM with sides in a ratio of 4: 1.a. What is the ratio of the areas of the tworectangles?b. Suppose the dimension of each rectangle is tripled.What is the new ratio of the sides of the rectangles?c. What is the ratio of the areas of these largerrectangles?d. Suppose only one pair of correspondingdimensions of each rectangle is doubled. What is thenew ratio of the sides of the rectangles?

SOLUTION:

a. The area of rectangle involves squaring therelationship of the width to the length , therefore theratio would be . b. If the dimensions of each side were tripled, wewould have a resulting ratio 4(3): 1(3) = 12: 3,which results in a ratio of 4:1. There is no change inthe ratio through multiplying each dimension by thesame number; the ratio of the sides remains thesame. c. The area of rectangle involves squaring therelationship of the width to the length , therefore itsratio would be , which reduces to16:1. d. If only one pair of corresponding dimensions ofeach side were doubled, we would have a resultingratio 4(2): 1(2) = 8: 2, which results in a ratio of 4:1.There is no change in the ratio through multiplyingeach dimension by the same number; the ratio of thecorresponding sides remains the same.

ANSWER:

a. 16:1b. 4:1c. 16:1d. 4:1

32. CHANGING DIMENSIONS In the figure shown,

a. Show that the perimeters of and have the same ratio as their corresponding sides.b. If 6 units are added to the lengths of each side, arethe new triangles similar? Explain.

SOLUTION:

a. The ratio of the corresponding sides of these two

right triangles are and the ratio of

their perimeters is . When each ratio

written in its most simplified form, you get .

b.

, which does not

equal a ratio of 1:3. Therefore, no, the sides are nolonger proportional.

ANSWER:

a.

b. No; the sides are no longer proportional.

33. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate similarity in squares.a. GEOMETRIC Draw three different-sizedsquares. Label them ABCD, PQRS, and WXYZ.Measure and label each square with its side length.b. TABULAR Calculate and record in a table theratios of corresponding sides for each pair of squares:ABCD and PQRS, PQRS and WXYZ, and WXYZ andABCD. Is each pair of squares similar?c. VERBAL Make a conjecture about the similarity



of all squares.

SOLUTION:

a. Choose three different side lengths for thesketches of your three squares. It would be a goodidea to use some decimals, as well as whole numberside lengths as options. Label them as directed. Sample answer:

b. When filling in this table, remember that all sides ofa square are congruent to each other. What do youobserve from your calculations?

ABCD is similar to PQRS; PQRS is similar to WXYZ;WXYZ is similar to ABCD. c. Sample answer: Since all the ratios ofcorresponding sides of squares are the same, allsquares are similar.

ANSWER:

a. Sample answer:

b.

ABCD is similar to PQRS; PQRS is similar to WXYZ;WXYZ is similar to ABCD.c. Sample answer: All squares are similar.



34. CHALLENGE For what value(s) of x is

SOLUTION:

Use the corresponding side lengths to write aproportion. Recall that the opposite sides in arectangle are congruent.

Solve for x.

ANSWER:

4

35. OPEN-ENDED Find a counterexample for thefollowing statement.All rectangles are similar.

SOLUTION:

Not all rectangles are similar.Shown are two rectangles. However, the ratios oftheir corresponding sides are not the same.Therefore, they cannot be similar.

ANSWER:

Sample answer:

36. REASONING Draw two regular pentagons of

different sizes. Are the pentagons similar? Will anytwo regular polygons with the same number of sidesbe similar? Explain.

SOLUTION:

When you draw two regular pentagons (5-sidedpolygons) recall that all sides of the same pentagonare congruent to each other and all angles arecongruent to each other as well. To determine oneangle measure of any regular pentagon, you can usethe sum of the interior angles of a polygon formula180(n-2), where n represents the number of sides,and substitute in 5 for n

Divide 540 degrees by the 5 and determine themeasure of one angle in a regular pentagon:

degrees

Thus, all the pentagons similar. And any two regularpolygons with the same number of sides be similar. The pentagons are similar because theircorresponding angles are congruent and theircorresponding sides are proportional. All of the anglesand sides in a regular polygon are congruent. Theangles will be congruent regardless of the size of thefigure, and since all of the sides are congruent theratios of the sides of one regular figure to a second



regular figure with the same number of sides will allbe the same. Therefore, all regular polygons with thesame number of sides are congruent.

ANSWER:

Yes; yes; sample answer: The pentagons are similarbecause their corresponding angles are congruent andtheir corresponding sides are proportional. All of theangles and sides in a regular polygon are congruent.The angles will be congruent regardless of the size ofthe figure, and since all of the sides are congruent theratios of the sides of one regular figure to a secondregular figure with the same number of sides will allbe the same. Therefore, all regular polygons with thesame number of sides are congruent.

37. WRITING IN MATH How can you describe therelationship between two figures?

SOLUTION: Sample answer: The figures could be described ascongruent if they are the same size and shape, similarif their corresponding angles are congruent and theircorresponding sides are proportional, and equal if theyare the same exact figure.

ANSWER: Sample answer: The figures could be described ascongruent if they are the same size and shape, similarif their corresponding angles are congruent and theircorresponding sides are proportional, and equal if theyare the same exact figure.

38. In the figure, JKLM ~ PQRS. What is the perimeterof PQRS?

A 18B 39.3C 50D 88.5

SOLUTION:

The perimeter of JKLM is 12 + 10 + 17 + 20 = 59. Because PS = 1.5(JM), the perimeter of PQRS is(1.5)59 = 88.5

ANSWER: D



39. In the figure, .

What is the perimeter of ?

SOLUTION: If the triangles are similar, then the correspondingsides are proportional to each other. Set up a proportion of corresponding sides to find DEand DF.

The perimeter of is .

ANSWER: 37.5

40. Braden drew two rectangles, RSTU and VWXY, sothat RSTU ~ VWXY. The ratio of the perimeter ofRSTU to the perimeter of VWXY is . Given that thelength of RSTU is 24 and the width of RSTU is 12,what is the length of VWXY?

A 9B 16C 18D 32

SOLUTION: According to Theorem 7.1 Perimeters of SimilarPolygons, if two polygons are similar, then theirperimeters are proportional to the scale factorbetween them. Therefore, if RSTU ~ VWXY, thecorresponding sides have the same ratio as the ratio

of the perimeters, which is . Sketch and label rectangles RSTU and VWXY.

The length of RSTU ( ) corresponds to the lengthof VWXY ( ). Set up a proportion of thecorresponding side lengths to the ratio of theperimeters.

The correct choice is D.

ANSWER: D



41. Two similar rectangles have a scale factor of 3 : 5.The perimeter of the larger rectangle is 65 meters.What is the perimeter in meters of the smallerrectangle?

SOLUTION: If two rectangles are similar and their scale factor is3:5, then their perimeters have the same scale factor.Set up a proportion comparing the perimeters of thetwo rectangles to the scale factor.

The perimeter of the smaller rectangle, in meters, is39.

ANSWER: 39

42. An architect is designing two triangular supportstructures for a roof. The support structures must besimilar. The architect designs them so that KLM ~ PQR, as shown. She wants the ratio of the

perimeter of KLM to the perimeter of PQR to be

.

What length should the architect use for ?

A 54 ftB 36 ftC 24 ftD 16 ft

SOLUTION: If the architect wants the perimeters of the triangles

to have the ratio of , then the corresponding sidesmust have the same ratio. To find QR, set up aproportion with its corresponding side from ΔKLM.

The correct choice is C.

ANSWER: C

43. MULTI-STEP Yasmina uses a coordinate plane todesign jewelry to sell at a craft fair. The figure showsa design for a pair of earrings that will be based onpolygon DEFG.



a. Yasmina decides to base the earrings on polygonHJKL instead. Use transformations to explain whypolygons DEFG and HJKL are similar. b. Write a similarity statement for the polygons. c. Write a proportion that shows how the sides of thepolygons are related. d. Suppose the perimeter of polygon DEFG is 6.2centimeters. What is the perimeter of polygon HJKL?Explain how you know.

SOLUTION: a. Map DEFG to HJKL using a dilation centered atthe origin with scale factor 0.5 followed by atranslation along <6, 5>. b. DEFG ~ HJKL, because this is the order ofcorresponding angles and sides in the two figures. c. The proportions should have corresponding sides inratios.

d. 3.1 cm; because the scale factor of the dilation is0.5, the perimeter of HJKL is 0.5 times the perimeterof DEFG.

ANSWER: a. Map DEFG to HJKL using a dilation centered atthe origin with scale factor 0.5 followed by atranslation along <6, 5>. b. DEFG ~ HJKL

c. d. 3.1 cm; because the scale factor of the dilation is0.5, the perimeter of HJKL is 0.5 times the perimeterof DEFG.



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