Chapter 7 – Similarity
7.1
Answer Key
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.
54, 54, 72
8.
9, 12, 15
9.
12 and 20
10.
64 and 112
11.
20 and 36
12.
12 and 44
13.
7.96 and 15.72
14.
1: 2
15.
1: 1
CK-12 Geometry Concepts
1
Chapter 7 – Similarity
7.2
Answer Key
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.
1 3 cups water
10.
60 marshmallows; 6 cups miniatures
11.
False
12.
True
13.
False
14.
False
15.
28
16.
18
17.
7
18.
24
19.
Answers will vary. Use the hint to simplify the proportion.
20.
Answers will vary. Use the hint to simplify the proportion.
2
CK-12 Geometry Concepts
2
Answer Key
Chapter 7 – Similarity
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, 𝐻𝐴 = 𝐴𝑇 = 𝐻𝑇
10.
3 5
11.
HT = 35
12.
IG = 27
13.
𝐵𝐼
𝐼𝐺
𝐵𝐺
3 5
14.
mE = 113, mQ = 112
15.
3 2
16.
BC = 12
17.
CD = 21
CK-12 Geometry Concepts
3
Answer Key
Chapter 7 – Similarity 18.
NP = 6
19.
No, 26 ≠ 12
20.
Yes, ∆𝐴𝐵𝐶~∆𝑁𝑀𝐿
21.
Yes, 𝐴𝐵𝐶𝐷~𝑆𝑇𝑈𝑉
22.
Yes, ∆𝐸𝐹𝐺~∆𝐿𝑀𝑁
23.
x = 12, y = 15
24.
31
25.
x = 20, y = 7
26.
x ≈14.6
27.
a ≈ 7.4, b = 9.6
28.
x = 6, y = 10.5
29.
121
30.
1:3
31.
30𝑢2 , 270𝑢2 , 1: 9, this is the ratio of the lengths squared or (3) .
32
18
CK-12 Geometry Concepts
1 2
4
Answer Key
Chapter 7 – Similarity
7.4
AA Similarity
Answers 1.
∆𝑆𝐴𝑀~∆𝑇𝑅𝐼
2.
𝑆𝐴 𝑇𝑅
3.
SM = 12
4.
TR = 6
5.
9 6
=
=
𝑆𝑀 𝑇𝐼
=
𝐴𝑀 𝑅𝐼
12 8
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.
Only two angles are needed because of the 3rd Angle Theorem.
11.
Congruent triangles have the same shape AND size. Similar triangles only have the same shape. Also, congruent triangles are always similar, but similar triangles are not always congruent.
12.
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.
13.
FE
14.
16
𝐴𝐸 𝐶𝐸
=
𝐵𝐸 𝐷𝐸
=
𝐴𝐵 𝐷𝐶
3 k 4
CK-12 Geometry Concepts
5
Answer Key
Chapter 7 – Similarity 15.
16.
If an acute angle of a right triangle is congruent to an acute angle in another right triangle, then the two triangles are similar.
E
D
D
G F
E GF
D
17.
𝑚∠1 + 𝑚∠2 = 90°, therefore 𝑚∠𝐺𝐷𝐹 = 𝑚∠2 and 𝑚∠𝐸𝐷𝐹 = ∠1. This shows that the three angles in each triangle are congruent to the three corresponding angles in each of the other triangles. Thus, they are all similar.
18.
DF
19.
GD
20.
FE
CK-12 Geometry Concepts
6
Chapter 7 – Similarity
7.5
Answer Key
Indirect Measurement
Answers 1.
13,000 ft
2.
97 ft
3.
19,400 ft
4.
70 ft
5.
29’ 2”
6.
24 ft
7.
12 ft
8.
Priya, she has the longer shadow.
9.
12’ 5.5”
10.
2’ 8”
11.
24 ft
12.
67’ 6”
13.
33’ 3”
14.
9’ 5. 3 ”
15.
Answers will vary.
CK-12 Geometry Concepts
7
Chapter 7 – Similarity
7.6
Answer Key
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: 7 15.4
10
12
= 22 = 26.4. 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.
𝐴𝐵 𝐷𝐹
8.
DH = 7.5
9.
Perimeter of ∆ABC = 36
𝐵𝐶
𝐴𝐶
= 𝐸𝐹 = 𝐷𝐹
Perimeter of ∆DEF = 27 The ratio is 4: 3.
10.
∆𝐴𝐵𝐶~∆𝐷𝐵𝐸
CK-12 Geometry Concepts
8
Answer Key
Chapter 7 – Similarity 11.
The triangles share B and
21 28
=
24 , 32
meaning that the two sides around B are
proportional. This is SAS Similarity (in the next concept). 12.
ED = 27
13.
𝐵𝐷 𝐴𝐵
14.
7 8 ̅̅̅̅ . Yes, 21 = 24. This proportion will be valid as long as ̅̅̅̅ 𝐴𝐶 ||𝐷𝐸
15.
No, 21 ≠ 27.
16.
x = 6, y = 3.5
𝐸𝐵
𝐷𝐸
= 𝐵𝐶 = 𝐴𝐶
7
36
CK-12 Geometry Concepts
9
Answer Key
Chapter 7 – Similarity
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 18 = 24. 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 ~ NOM
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.
27
CK-12 Geometry Concepts
36
10
Chapter 7 – Similarity
7.8
Answer Key
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
14.
1.5b
15.
1: 1
CK-12 Geometry Concepts
11
Chapter 7 – Similarity
7.9
Answer Key
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
16.
a = 42 m, b = 56 m
CK-12 Geometry Concepts
12
Answer Key
Chapter 7 – Similarity
7.10
Proportions with Angle Bisectors
Answers 1.
8.38
2.
3.4
3.
a = 1.8, b =
4.
x = 5, y = 7
5.
3.2a
6.
Casey mistakenly used the length of the angle bisector in the proportion rather than the
7 3
5
other side length. The correct proportion is 𝑎 =
7.
5
8.
1
9.
0.75
10.
1.38
11.
2.14
12.
2
13.
−2, 2
14.
0, 2
15.
−1, 3
16.
1.09
17.
13.125
18.
7.4375
19.
3.2
CK-12 Geometry Concepts
7 5
, thus 𝑎 =
25 . 7
13
Answer Key
Chapter 7 – Similarity
7.11
Dilation
Answers 1.
20, 26, 34
2.
2 3, 3, 5
3.
𝑘 = 11
4.
original: 20, dilation: 80, ratio: 4: 1
5.
If k = 1, then the dilation is congruent to the original figure.
2
14
6.
7.
𝑫′𝑶′ 𝑫𝑶
= 𝟑,
𝑶′𝑮′ 𝑶𝑮
= 𝟑 and
𝑮′𝑫′ 𝑮𝑫
= 𝟑.
8.
3
9.
To dilate the original figure by a scale factor of 4 make one additional tick mark with your compass.
10.
𝟏 To dilate the original figure by a scale factor of 𝟐, find the midpoints of ̅̅̅̅ 𝑪𝑮, ̅̅̅̅ 𝑪𝑶 and ̅̅̅̅ 𝑪𝑫.
CK-12 Geometry Concepts
14
Chapter 7 – Similarity 11.
3: 2
12.
9: 4
13.
9, 21, 27
14.
5 by 6
15.
1: 4
CK-12 Geometry Concepts
Answer Key
15
Chapter 7 – Similarity
7.12
Answer Key
Dilation in the Coordinate Plane
Answers 3
1.
𝑘=2
2.
𝑘=9
3.
𝑘=2
4.
A’(6, 12), B’(-9, 21), C’(-3, -6)
5.
A’(9, 6), B’(-3, -12), C’(0, -7.5)
1
CK-12 Geometry Concepts
16
Chapter 7 – Similarity 6.
Answer shown in black triangle below.
7.
k = 2. (Plots shown in red triangle below)
8.
A’’(4, 8), B’’(48, 12), C’’(40, 40). (Plots shown in blue triangle below.
9.
k=2
10.
𝑂𝐴 ≈ 2.34, 𝐴𝐴′ ≈ 2.24, 𝐴𝐴′′ ≈ 6.71, 𝑂𝐴′ ≈ 4.68, 𝑂𝐴′′ ≈ 13.42
11.
𝐴𝐵 ≈ 11.18, 𝐴′ 𝐵′ ≈ 22.36, 𝐴′′ 𝐵′′ ≈ 44.72
12.
a)
Answer Key
OA: OA’ = 1: 2, AB: A’B’ = 1: 2 These ratios are the same because this is the value of the scale factor.
b)
OA: OA’’ = 1: 4, AB: A’’B’’ = 1: 4
These ratios are the same because this is the value of the scale factor.
CK-12 Geometry Concepts
17
Chapter 7 – Similarity
Answer Key
13.
14.
x=3
15.
y = 2x + 1
16.
(3, 7)
17.
This point is the center of the dilation.
18.
The scale factor is 3.
CK-12 Geometry Concepts
18
Answer Key
Chapter 7 – Similarity
7.13
Self-Similarity
Answers 1.
Erase the middle third of each line.
2-10.
Stage 0 Stage 1
Number of Segments 1 2
Stage 2
4
Stage 3
8
Stage 4
16
Stage 5
32
11.
There will be 2𝑛 segments.
12.
Each segment will be 3𝑛 units.
Length of each Segment 1 1 3 1 9 1 27 1 81 1 243
Total Length of the Segments 1 2 3 4 9 8 27 16 81 32 243
1
13.
CK-12 Geometry Concepts
19
Answer Key
Chapter 7 – Similarity
14.
Color No Color 15.
Stage 0 0 1
Stage 1 1 8
Stage 2 9 64
Stage 3 73 512
Answers will vary.
CK-12 Geometry Concepts
20
