7.1 Forms of Ratios

Chapter 7 – Similarity Answer Key

CK-‐12 Basic Geometry Concepts 1

7.1 Forms of Ratios

Answers

1. a) 4: 3

b) 5: 8

c) 6: 19

d) 6: 8: 5

2. 1: 1

3. 1: 2

4. 2: 1

5. 1: 1

6. 5: 4: 3

7. 512

8. 11

9. 1930

10. 54° and 72°

11. 12 and 20

12. 64 and 112

13. 20

14. 240

15. 30



7.2 Proportion Properties

Answers

1. x = 12

2. x = −5

3. y = 16

4. x = 12, −12

5. y = −21

6. z = 3.75

7. 13.9 gal

8. President = $800,000, VP = $600,000, Financial Officer = $400,000

9. False

10. True

11. False

12. False

13. 28

14. 18

15. 7

16. 24



7.3 Similar Polygons and Scale Factors

Answers

1. True; all the angles are equal for all equilateral triangles. All the sides are congruent in every equilateral triangle, so the proportion of the sides is the scale factor.

2. False; the ratio of the bases can be different than the ratio of the legs.

3. False; the ratio of the lengths can be different than the ratio of the widths.

4. False; the angles of every rhombus do not have to be equal.

5. True; same reasoning as an equilateral triangle. All regular polygons are similar.

6. True; if two polygons are congruent, then they are also similar. The scale factor would be 1:1.

7. False; this is the converse of #6. Similar polygons can have a scale factor other than 1:1, meaning they would not be congruent.

8. True; all regular polygons are similar.

9. ∠B ≅ ∠H, ∠I ≅ ∠A, ∠G ≅ ∠T, !"!"

= !"!"= !"

!"

10. !!

11. HT = 35



12. IG = 27

13. Ratio is !!

14. The two courts are not similar because they do not reduce to the same ratio.

15. 16: 9 ≠ 4: 3, these ratios are not the same, so TV ratios are not the same.

16. m∠E = 113°, m∠Q = 112°

17. !!

18. BC = 12

19. CD = 21

20. NP = 6

21. No, !"!"≠ !"

!"

22. Yes, ∆𝐴𝐵𝐶~∆𝑁𝑀𝐿

23. Yes, 𝐴𝐵𝐶𝐷~𝑆𝑇𝑈𝑉

24. Yes, ∆𝐸𝐹𝐺~∆𝐿𝑀𝑁

25. Yes, 𝑄𝑅𝑆𝑇~𝐵𝐶𝐷𝐴

26. No, 𝑚∠𝑀 ≠ 𝑚∠𝐴 and 𝑚∠𝑁 ≠ 𝑚∠𝐶

27. No, !!"≠ !

!"

28. Yes, ∆𝐸𝐹𝐺~∆𝑀𝐿𝑁

29. Yes, 𝐴𝐵𝐷𝐶~𝐸𝐹𝐺𝐻

30. No, we do not know any angle measures.



7.4 AA Similarity

Answers

1. ∆𝑆𝐴𝑀~∆𝑇𝑅𝐼

2. !"!"= !"

!"= !"

!"

3. SM = 12

4. TR = 6

5. !!= !"

!

6. ∆𝐴𝐵𝐸~∆𝐶𝐷𝐸 because ∠BAE ≅ ∠DCE and ∠ABE ≅ ∠CDE by the Alternate Interior Angles Theorem. There is not enough information to say another other triangles are similar.

7. Possible Answers !"!"= !"

!"= !"

!"

8. Possible Answers ∆AED and ∆BEC, ∆AEB and ∆BEC, ∆ABE and ∆ABC, ∆ECD and ∆AED

9. AC = 22.4

10. Yes, right angles are congruent and solving for the missing angle in each triangle, we find that the other two angles are congruent as well.

11. 34

FE k=

12. 16

13. If an acute angle of a right triangle is congruent to an acute angle in another right triangle, then the two triangles are similar.



14. Congruent triangles have the same shape AND size. Similar triangles only have the same shape. Congruent triangles are always similar. Similar triangles are not always congruent.

15. No only vertical angles are congruent. One angle is not enough to say the triangles are similar.

16. Yes, ∆𝐿𝑁𝐾~∆𝐽𝑁𝑀.

17. Yes, m∠IFG = 105°, ∆𝐹𝐼𝐻~∆𝐺𝐼𝐹.

18. Yes, 𝐸𝐵||𝐷𝐶, so all the angles are congruent; ∆𝐴𝐸𝐵~∆𝐴𝐷𝐶.

19. No, there are no congruent angles.

20. Yes, vertical angles are congruent and the 55° angles are congruent; ∆𝑇𝑈𝑊~∆𝑋𝑈𝑉.

21. No, 𝐸𝐺 ∦ 𝐷𝐶.



7.5 Indirect Measurement

Answers

1. 13,000 ft

2. 97 ft

3. 19,400 ft

4. 12 ft

5. Karen, she has the longer shadow.

6. 12’ 5.5”

7. 2’ 8”

8. 24 ft

9. 67’ 6”

10. 33’ 3”



7.6 SSS Similarity

Answers

1. If all three sides in one triangle are proportional to the three sides in another, then the two triangles are similar.

2. Two triangles are similar if the corresponding sides are proportional.

3. Yes, by SSS. There are 2.2 cm in an inch, so if we were to put the larger triangle into centimeters the sides would be 15.4, 22.0, and 26.4. Writing the proportions we have:

!!".!

= !"!!= !"

!".!. Therefore, the side lengths are proportional.

4. No. In #3, we converted the larger triangle into centimeters. From these measurements, we can see that the larger triangle is about double the size of the smaller triangle.

5. There are 2.2 cm in an inch, so that is the scale factor.

6. ∆𝐴𝐵𝐶~∆𝐷𝐹𝐸

7. !"!"= !"

!"= !"

!"

8. DH = 7.5

9. Perimeter of ∆ABC = 36

Perimeter of ∆DEF = 27

The ratio is 4: 3.



10. ∆𝐴𝐵𝐶~∆𝐷𝐵𝐸

11. The triangles share ∠B and !"!"= !"

!", meaning that the two sides around ∠B are

proportional. This is SAS Similarity (in the next concept).

12. ED = 27

13. !"!!= !"

!"= !"

!"

14. Yes, !!"= !

!". This proportion will be valid as long as 𝐴𝐶||𝐷𝐸.

15. No, !!"≠ !"

!".

16. x = 6, y = 3.5



7.7 SAS Similarity

Answers

1. If two sides in one triangle are proportional to two sides in another and the corresponding angles are congruent, then the triangles are similar.

2. Yes, ∠ABE ≅ ∠CBD and !"!"= !"

!". By SAS, ∆𝐴𝐵𝐸~∆𝐷𝐵𝐶.

3. x = 3

4. x = 2

5. x = 5

6. Yes (we don’t know which angles are what measurement, so similarity statements will vary).

7. No

8. Yes, ~NQP NOMΔ Δ

9. No

10. No

11. Yes, cannot write a similarity statement because the vertices are not labeled.

12. No, we do not know if the lines are parallel. Cannot assume any angles are congruent.

13. No, sides don’t line up.

14. No

15. Yes, cannot write a similarity statement because the vertices are not labeled.



7.8 Triangle Proportionality

Answers

1. 14.4

2. 21.6

3. 16.8

4. 45

5. 2: 3

6. 3: 5

7. 2: 3 is the ratio of the segments created by the parallel lines, 3: 5 is the ratio of the similar triangles.

8. Yes

9. No

10. Yes

11. No

12. Yes

13. No



7.9 Parallel Lines and Transversals

Answers

1. b = 12.8

2. y = 3

3. x = 4

4. a = 4.8, b = 9.6

5. a = 4.5, b = 4, c = 10

6. 3072

7. 576

8. 4608

9. 2.625

10. 3

11. 0.5

12. 12.5

13. 15.625

14. one-third of c

15. half of d



7.10 Proportions with Angle Bisectors

Answers

1. 8.38

2. 3.4

3. 5

4. 1

5. 0.75

6. 1.38

7. 2.14

8. 2

9. −2, 2

10. 0, 2

11. −1, 3

12. 1.09

13. 13.125

14. 7.4375

15. 3.2



7.11 Dilation

Answers

1. 2.

10

828

35

15

1122

30

3. 4.

13.5

1216

18

6

8

15

20

5. 20, 26, 34

6. 2 !!, 3, 5

7. 7.5, 10, 12.5

8. 2, 3, 4

9. 𝑘 = !"!!

10. 𝑘 = !!

11. 𝑘 = !!



7.12 Dilation in the Coordinate Plane

Answers

1. 𝑘 = !!

2. 𝑘 = 9

3. 𝑘 = !!

4. A’(6, 12), B’(-9, 21), C’(-3, -6)

5. A’(9, 6), B’(-3, -12), C’(0, -7.5)



6. Black triangle in graph below.

7. k = 2 Red triangle in graph below

8. Blue triangle in graph below. A’’(4, 8), B’’(48, 12), C’’(40, 40)

9. k = 2



10. 𝑂𝐴 ≈ 2.34

11. 𝐴𝐴′ ≈ 2.24

12. 𝐴𝐴!! ≈ 6.71

13. 𝑂𝐴! ≈ 4.68

14. 𝑂𝐴!! ≈ 13.42

15. 𝐴𝐵 ≈ 11.18

16. 𝐴!𝐵! ≈ 22.36

17. 𝐴!!𝐵!! ≈ 44.72

18. OA: OA’ = 1: 2, AB: A’B’ = 1: 2

These ratios are the same because this is the value of the scale factor.

19. OA: OA’’ = 1: 4, AB: A’’B’’ = 1: 4

These ratios are the same because this is the value of the scale factor.



7.13 Self-Similarity

Answers

1. Erase the middle third of each line.

2.

Number of Segments

Length of each Segment

Total Length of the Segments

Stage 0 1 1 1 Stage 1 2 1

3

23

Stage 2 4 1

9

49

Stage 3 8 1

27

827

Stage 4 16 1

81

1681

Stage 5 32 1

243

32243

3. There will be 2! segments.

4.



5.

6.

Stage 0 Stage 1 Stage 2 Stage 3 Color 0 1 9 73 No Color 1 8 64 512

7. Possible Answers Many different flowers (roses) and vegetables (broccoli and cauliflower) are examples of fractals in nature.

Documents

7.1 Forms of Ratios