Similar Figures Chapter Problems
Ratios and Proportions
Class work
Simplify the ratio.
1. 15 in to 45 in
2. 27yd to 6yd
3. 12 days to 4 weeks
4. 6 years to 1 decade
Solve the proportion
5.
6.
7.
8.
x-2 1
=
x -1 2
Tell whether the statement is true or false.
9. If
, then
10. If
, then
11. If
, then
12. The scale on the blueprint of a house is 0.04in = 1 foot. If the width of the kitchen on the
blueprint is 1 inch, what is the actual width of the kitchen?
13. There are 350 people at the school basketball game. The ratio of the students to adults is 6:1.
How many students attended the game?
Homework
Simplify the ratio.
14. 40 feet to 12 feet
15. 8 days to 14 days
16. 150 feet to 1 mile
17. 20 ounces to 3 pounds
Solve the proportion.
18.
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19.
20.
21.
Tell whether the statement is true or false.
22. If
, then
23. If
, then
24. If
, then
25. Mike, Angela, and Victor have $160 in a ratio of 7:5:4. How much do they each have?
26. You made a 3-foot model of your home, using a scale of 1:42. What is the actual height of your
home?
Similar Polygons using Transformations
Classwork
Use the definition of similarity in terms of similarity transformations to determine if the two figures
are similar. If similar, give the transformations in coordinate notation.
27.
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28.
29.
30.
31.
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Homework
32.
33.
34.
35.
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36.
Similar Polygons using Corresponding Parts
Classwork
37. Given that triangle XYZ ~ triangle LMN.
a. Write as many congruence statements as possible about the sides and / or angles.
b. Write the statement of proportionality.
c. Write 5 more similarity statements.
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38. The polygons below are similar.
a. Write a similarity statement.
b. What is the similarity ratio?
c. What is the scale factor?
d. List all congruent angles.
e. Write the statement of proportionality.
39. Decide whether the polygons are similar.
a. If yes, write a similarity statement.
b. What is the similarity ratio?
c. What is the scale factor?
In problems 40-41 DEFGHIJK, solve for the variables.
40.
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41.
Homework
42. Given that triangle PQR ~ triangle DEF.
a. Write as many congruence statements as possible about the sides and / or angles.
b. Write the statement of proportionality.
c. Write 5 more similarity statements.
43. The polygons below are similar.
a. Write a similarity statement.
b. What is the similarity ratio?
c. What is the scale factor?
d. List all congruent angles.
e. Write the statement of proportionality.
G
F
B
108°
35°
C
A
35°
135°
108°
D
135°
E
J
I
In problems 44-45, decide whether the polygons are similar.
a. If yes, write a similarity statement.
b. What is the similarity ratio?
c. What is the scale factor?
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H
44.
45.
46. ABCDPQRS. Solve for the variables.
Similar Triangles
Classwork
47. Determine if the triangles are similar. If so, state the similarity postulate or theorem.
a.
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b.
c.
48. Given: DH || FG
Prove: DDEH : DGEF
F
D
E
H
G
49. Given: ÐR @ ÐA , PR = 9 , QR = 7.2 ,
Prove: DQPR : DBCA
Geometry-Similar Figures
AB = 4.8 , and AC = 6
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ÐA @ ÐD , ÐB @ ÐE 
Prove: DABC : DDEF using similarity transformations
50. Given:
E
B
C
A
51.
Given:
Prove:
F
D
AB BC CA


DE EF FD
DABC : DDEF using similarity transformations
E
B
C
A
F
D
Homework
52. Determine if the triangles are similar. If so, state the similarity postulate or theorem.
a.
b.
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c.
53. Given: BD || AE
Prove: DACE : DBCD
54. Given: mÐP = 48°, mÐQ = 55° ,
Prove: DPQR : DABC
55. Given:
Prove:
mÐB = 55° , mÐC = 77°
AB CA

, ÐA @ ÐD
DE FD
DABC : DDEF using similarity transformations
E
B
A
C
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F
D
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Proportions in Similar Triangles
Classwork
In problems 56-58, determine if DE || BC .
56.
57.
58.
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Solve for y.
59.
60.
61.
62.
63.
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64. ABC is mapped to ADE under a dilation with a scale factor of 3, explain why
BC is parallel to DE .
6
D
4
E
2
B
C
A
5
65. Prove the Side Splitter Theorem.
Given BD ║ AE
Prove
CB CD

CA CE
C
B
A
D
E
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Homework
In problems 66-68, determine if DE || BC .
66.
67.
68.
Solve for y.
69.
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70.
71.
72.
73.
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74. ABC is mapped to ADE under a dilation with a scale factor of 1.5, explain why BC is parallel to DE .
6
D
4
B
E
2
C
A
5
75. Prove the Converse to the Side Splitter Theorem.
Given
AB ED

BC DC
C
Prove BD ║ AE
B
A
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D
E
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Similar Circles
Class work
76. Describe the similarity transformations needed to map circle A to circle A’. Point A is the center
of the dilation.
a. Find the constant of dilation.
b. Identify the translation vector.
77. Which similarity transformations can map circle A with center (0,0) and radius 2 to circle B with
center (-2, 3) and radius 4. Point A is the center of the dilation.
78. Which similarity transformations can map circle A with center (-3, -5) and radius 2 to circle B
with center (-6, 7) and radius 3. Point A is the center of the dilation.
79. Which similarity transformations can map circle A with center (4,7) and radius 8 to circle B with
center (-2, 10) and radius 4. Point A is the center of the dilation.
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80. Prove all circles are similar.
Given circle A with radius x and circle B with radius y
Prove circle A is similar to circle B
A
y
x
B
Homework
81. Describe the similarity transformations needed to map circle A to circle A’. Point A is the center
of the dilation.
a. Find the constant of dilation.
b. Identify the translation vector.
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82. Describe the similarity transformations needed to map circle A with radius AB to circle A with
radius AB' . Point A is the center of the dilation.
a. Find the constant of dilation.
b. Identify the translation vector.
83. Which similarity transformations can map circle A with center ( 3,3) and radius 5 to circle B with
center (-3, 3) and radius 4. Point A is the center of the dilation.
84. Which similarity transformations can map circle A with center (-3, -3) and radius 4 to circle B
with center (-3, -3) and radius 5. Point A is the center of the dilation.
85. Which similarity transformations can map circle A with center (4,7) and radius 6 to circle B with
center (-2, 10) and radius 6. Point A is the center of the dilation.
Solve Problems using Similarity
Class work
86. A basketball hoop in your backyard casts a shadow 109 inches long. You are 5 feet 8 inches tall
and cast a shadow 62 inches long. Find the height of the basketball hoop in inches. Round your
answer to the nearest whole number.
87. You want to know the approximate height of a very tall pine tree. You place a mirror on the
ground and stand where you can see the top of the tree in the mirror. How tall is the tree? The
mirror is 24 feet from the base of the tree. You are 24 inches from the mirror and your eyes are
6 feet above the ground. Round your answer to the nearest tenth.
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88. To find the distance d across a lake, you locate the points as shown. Find d. Round your answer
to the nearest tenth.
120 ft
d
15 ft
20 ft
89. A graphic designer wants to design a new grid system for a poster. The poster is 27 inches by 36
inches. The grid must have margins of 2 inch along all edges. There must be 4 rows of
rectangles. The rectangles must be similar in size to the poster.
a. What should be the height of the rectangles?
b. What should be the width of the rectangles?
c. How many columns of rectangles can there be?
Homework
90. A yardstick casts a shadow 1 ft long. A nearby tree casts a 16 ft shadow. How tall is the tree?
Round your answer to the nearest tenth.
91. You want to know the approximate height of a tall oak tree. You place a mirror on the ground
and stand where you can see the top of the tree in the mirror. How tall is the tree? The mirror is
24 feet from the base of the tree. You are 36 inches from the mirror and your eyes are 5 feet
above the ground. Round your answer to the nearest tenth.
92. To find the distance d across a lake, you locate the points as shown. Find d. Round your answer
to the nearest tenth.
28 ft
15 ft
56 ft
d
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93. A graphic designer wants to design a new grid system for a poster. The poster is 27 inches by 36
inches. The grid must have margins of 1 inch along all edges. There must be 5 rows of
rectangles. The rectangles must be similar in size to the poster.
a. What should be the height of the rectangles?
b. What should be the width of the rectangles?
c. How many columns of rectangles can there be?
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Similar Figures Unit Review
Multiple Choice - Choose the correct answer for each question. No partial credit will be given.
1.
Simplify the ratio 15 inches to 3 inches.
a. 15 to 9
b. 1:5
c. 5/3
d. 5/1
2.
Solve the proportion.
3.
a. 30
b. 54
c. 24
d. 27
Solve the proportion.
a.
b.
c.
d.
x=5
x = -5
x=1
x = 0.5
4. Use the definition of similarity, C is the center of dilation.
10
D''
8
6
C'' D
B''
4
B
C
2
5
a. Not Similar
b. Translation (x,y) -> (x-1, y+3)
followed by constant of dilation=2
c. Constant of dilation=2
followed by Translation (x,y) -> (x-1, y+3)
d. b and c order doesn’t matter
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5.
Decide whether the triangles are similar. If so, write a similarity statement.
a.
b.
c.
d.
Yes, DABC : DDEF
Yes, DABC : DDFE
Yes, DABC : DFDE
The triangles are not similar
In problems 6-7, JKLMPQRS, Find x.
6.
a.
b.
c.
d.
4
6.67
2.5
3.75
a.
b.
c.
d.
55
114
56
135
7.
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Describe the similarity transformation needed to map ABC to A’B’C’ using coordinate
notation.
8.
C
B8
6
4
2
B'
C'
A
5
a. Not Similar
b. (x, y) -> (1x/4, 1y/4)
c. (x, y) -> (4x, 4y)
d. (x, y) -> (6x, 6y)
What is the similarity ratio r needed to map ABC to A’B’C’? What is the scale factor f?
9.
C
B8
6
4
2
B'
C'
A
5
a.
b.
c.
d.
r=4, f=4
r=1/4, f=1/4
r=4, f=1/4
r=1/4, f=4
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10. Determine if the triangles are similar. If so, state the similarity postulate or theorem.
a.
Yes, by AA
b.
Yes, by SSS
c.
d.
Yes, by SAS 
The triangles are not similar
11.
a. 3.6
b. 4.4
c. 40
d. 48
12. Solve for y
a.
b.
c.
d.
8
4.5
6
12
13. Which similarity transformations can map circle A with center (-8,-3) and radius 6 to circle B
with center (4,-2) and radius 3. Point A is the center of the dilation.
a. (x, y) -> (x + 12, y +1)
constant of dilation=2
b. (x, y) -> (x + 4, y + 1)
constant of dilation=1/2
c. (x, y) -> (x + 12, y - 1)
constant of dilation=1/2
d. None of the above
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Short Constructed Response - Write the answer for each question. No partial credit will be given.
14. The scale on a map of the US, 1 inch = 250 miles. New York is 12 inches from California. What is the actual
distance between the cities.
15. For the diagram below, ADE is mapped to ABC under a dilation with a scale factor of 1/3, explain why
BC is parallel to DE .
6
D
4
E
2
B
C
A
5
16. Your school casts a shadow 30 feet long. At the same time a person 6 feet casts a shadow 4 feet
long. Sketch and label a diagram. Find the height of your school. Round your answer to the nearest tenth.
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Extended Constructed Response - Solve the problem, showing all work. Partial credit may be given.
17. Given B  D, C  E
Prove triangle ABC ~ triangle ADE using similarity transformations
D
4
B
2
A
C
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5
E
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Answer Key
1. 1/3
2. 9/2
3. 3/7
4. 3/5
5. 2.67
6. 5.25
7. 5
8. 5
9. true
10. true
11. false
12. 25 feet
13. 300 students
14. 10/3
15. 4/7
16. 5/176
17. 5/12
18. 2
19. 3
20. 3
21. -1
22. true
23. false
24. true
25. Mike - $70, Angela $50, Victor - $40
26. 126 feet
27. (x,y) ->(x+4,y+1)
28. (x,y)->(x/2,y/2)
29. (x,y)->(2x,2y)
30. (x,y)->(x,-y)
(x,y)->((x/2,y/2)
31. not similar
32. (x,y)->(x/2,y/2)
33. (x,y)->(3x/2,3y/2)
34. not similar
35. (x,y)->((x/2,y/2)
(x,y)->(x,y+2.5))
36. (x,y)->(-y,x)
37.
a. ÐX @ ÐL , ÐY @ ÐM,
ÐZ @ ÐN
Geometry-Similar Figures
b. XY/LM=YZ/MN=XZ/LN
c. XZY~LNM,YXZ~MLN,
YZX~MNL,ZXY~NLM,
ZYX~NML
38.
a. ABCD~EFGH
b. r=2/3
c. f=3/2
d. <A=<E, <B=<F, <C=<G,
<D=<H
e. AB/EF=BC/FG=
CD/GH=DA/HE
39. not similar
40. w=12, x=28, y=2.4,
z=2.67
41. w=8, x=3, y=4.5
42.
a. ÐP @ ÐD, ÐQ @ ÐE,
ÐR @ ÐF
b. PQ/DE=QR/EF=PR/DF
c. PRQ~DFE, QPR~EDF,
QRP~EFD, RPQ~FDE,
RQP~FED
43.
a. ABCDE~HIJFG
b. not enough info
c. not enough info
d. <A=<H, <B=<I, <C=<J,
<D=<F, <E=<G
e.
AB/HI=BC/IJ=CD/JF=DE/FG
=EA/GH
44.
a. ABCD~QPSR
b. r=2/1
c. f=1/2
45. not similar
46. w=114,x=4.5,
y=2.25,z=87
47.
a. not similar
b. yes by SAS
~29~
c. not similar
48. <D = <G and <E = <E so by
AA
49. (QR/BA)=(PR/AC) and
<R=<A so by SAS
50. see below
51. see below
52.
a. yes, by AA or SAS
b. not similar
c. yes, by SSS
53. <B=<A and <D=<E so by
AA
54. <Q=<B and <R=<C so by
AA~
55. see below
56. yes
57. no
58. no
59. 12
60. 10.67
61. 7
62. 11.36
63. 8.75
64. ABC~ADE because a
dilation is a similarity
transformation.
<B=<D because
corresponding angles of ~
triangles are congruent.
BC is parallel to DE by the
corresponding angles
converse.
65. see below
66. yes
67. yes
68. no
69. 14
70. 10
71. 11.25
72. 4
73. 10
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74. ABC~ADE because a
dilation is a similarity
transformation.
<B=<D because
corresponding angles of ~
triangles are congruent.
BC is parallel to DE by the
corresponding angles
converse.
75. see below
76.
a. constant of dilation=1/3
b. vector AA’ = <7, -3>
77. constant of dilation=2
(x,y)->(x-2, y+3)
78. constant of
dilation=3/2
(x,y)->(x-3, y+12)
79. constant of
dilation=1/2
(x,y)->(x-6, y+3)
80. see below
81.
a. constant of dilation=3/2
(x,y) -> (x+3, y+5)
b. vector AB = <3, 5>
82.
a. constant of dilation=2
b. vector AA’ = <0, 0>
83. constant of
dilation=5/4
(x,y)->(x-6, y)
84. constant of
dilation=5/4
(x,y)->(x,y)
85. constant of dilation=1
(x,y)->(x-6, y+3)
86. 120 inches
87. 72 ft
88. 90 ft
89.
a. 8 in or 5.75 in
b. 6 in or 7.67 in
c. 3 or 4
90. 48 ft
91. 40 ft
92. 45 ft
93.
a. 6.8 in or 5 in
b. 5.1 in or 6.67 in
c. 4 or 5
1. D
2. C
3. A
4. D
5. C
6. D
7. A
8. B
9. C
10. C
11. A
12. A
13. D
14. 3000 miles
15. ADE~ABC because a
dilation is a similarity
transformation.
<D=<B because
corresponding angles of ~
triangles are congruent.
BC is parallel to DE by the
corresponding angles
converse.
16. 45 feet
17. see below
Unit Review
Proofs
50. AA~ proof using transformations
<A=<D,<B=<E
Dilate ABC with sf=DE/AB
ABC~A'B'C'
<A=<A',<B=<B'
A'B'=(DE/AB)*AB=DE
<A'=<D,<B'=<E
A'B'C'=DEF
A'B'C' ~ DEF
ABC ~ DEF
Given
Def of scale factor
Def of dilation
corr angles of ~ triangles are cong
simplify
Transitive Prop of
congruence
ASA congruence
Def of congruence
Transitive prop of
~
51. SSS~ proof using transformations
AB/DE=BC/EF=CA/FD
Geometry-Similar Figures
Given
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DE/AB=EF/BC=FD/CA
Dilate ABC with scale factor k =DE/AB
ABC~A'B'C'
A'B'=(DE/AB)(AB)=DE
B'C'=(EF/BC)(BC)=EF
C'A'=(FD/CA)(CA)=FD
A'B'C' @ DEF
A'B'C'~DEF
ABC~DEF
Definition of Proportions
Definition of Scale Factor
Definition of Dilation
Simplify
Substitution / Simplify
Substitution / Simplify
SSS=
Definition of @
Transitive Property of ~
55. SAS~ proof using transformations
AB/DE=CA/FD
<A=<D
DE/AB=FD/CA
Dilate ABC with scale factor k =DE/AB
ABC~A'B'C'
A'B'=(DE/AB)(AB)=DE
C'A'=(FD/CA)(CA)=FD
A'B'C' @ DEF
A'B'C'~DEF
ABC~DEF
Given
Given
Definition of Proportions
Definition of Scale Factor
Definition of Dilation
Simplify
Substition / Simplify
SAS @
Definition of @
Transitive Property of ~
65. Side Splitter Theorem Proof
EA parallel to DB
<CBD=<CAE
<C=<C
CBD ~ CAE
CA/CB=CE/CD
CB+BA=CA, CD+DE=CE
(CB+BA)/CB=(CD+DE)/CD
CB/CB+BA/CB=CD/CD+DE/CD
1+BA/CB=1+DE/CD
BA/CB=DE/CD
CB/BA=CD/DE
Given
corresponding angles postulate
reflexive prop of congruence
AA~
corr sides of ~ triangles are prop
segment addition postulate
substitution
simplify
simplify
subtraction prop of =
property of proportions
75. Converse of Side Splitter Theorem Proof
AB/BC=ED/DC
1+AB/BC=1+ED/DC
BC/BC+AB/BC=DC/DC+ED/DC
(BC+AB)/BC=(DC+ED)/DC
BC+AB=AC,DC+ED=CE
AC/BC=CE/DC
Geometry-Similar Figures
Given
Addition property of =
substitution
simplify
segment addition postulate
substitution
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<C=<C
BCD ~ ACE
<CBD = <CAE
BD is parallel to AE
reflexive property of congruence
SAS~
corresponding angles of ~ triangles are congruent
corresponding angles converse
80. Prove all circles are similar
Translate circle A with vector AB getting circle A'
circle A is congruent to circle A'
center of circle A' is B
Dilate circle A' with scale factor k = y/x
circle A' ~ circle B
circle A ~ circle B
Definition of Translation
Definition of Translation
Definition of Translation
Definition of Dilation
Definition of Dilation
Transitive Property of ~
Unit Review #17 AA~ proof using transformations
<B=<D,<C=<E
Dilate ABC with sf=DE/BC
ABC~A'B'C'
<B=<B',<C=<C'
B'C'=DE/BC*BC=DE
<B'=<D,<C'=<E
A'B'C'=ADE
A'B'C' ~ ADE
ABC ~ ADE
Geometry-Similar Figures
Given
Def of scale factor
Def of dilation
corr angles of ~ triangles are cong
simplify
Transitive Prop of
congruence
ASA congruence
Def of congruence
Transitive prop of
~
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