7.1 forms of ratios
TRANSCRIPT
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 2
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 3
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 4
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 5
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 6
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 7
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”
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 8
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 9
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 10
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 11
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 12
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 13
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 14
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 = !!
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 15
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)
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 16
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
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 17
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 18
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.
Chapter 7 – Similarity Answer Key
CK-‐12 Basic Geometry Concepts 19
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.