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I. Model Problems.
II. Practice Problems
III. Challenge questions
VI. Answer Key
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I. Model Problems: Ratios in Similar Polygons
Definition of Similar Polygons
Two Polygons are similar if and only if the corresponding angles are congruent
and the corresponding side lengths are proportional. The mathematical symbol
used to state that two polygon are similar is .
D
A
B
C
F
E
The Corresponding Angles are: A and D , B and E , and C and F .
The Corresponding Sides are: AB and DE , BC and EF , and CA and FD .
If we state that ABC DEF
then the following is true:
 A  D
 B  E
 C  F

AB BC CA


DE EF FD
Corresponding angles are congruent.
Corresponding Side lengths are proportional.
ABC  DEF
Notice the Corresponding Angles line up in the same order in the similarity
statement.
 The Similarity Ratio should be ratio between the corresponding sides
lengths of the triangles.
 Overall, the triangles are the same shape, meaning all angles have the same
measure, but one triangle is bigger than the other.
Example 1
For the two triangle, write a similarity statement, then determine what we can
conclude based on the figures. What is the similarity ratio between the two
triangles?
J
8
I
K
3
5
6
U
4
10
S
P
Answer: The similarity statement is KUS PJI based on the angles that are
congruent. Additionally we can make the following conclusions:
K  P, U  J , S  I based corresponding angles being congruent.
KU US KS


PJ
JI
PI
based on corresponding sides being proportional.
To find the similarity ratio, insert the side lengths.
3 4 5 1
   , so the similarity ratio is 1:2.
6 8 10 2
Example 2
Find the measure all missing sides and angles using the properties of similar
polygons. What is the similarity ratio? WZYX EFDB
8
Answer: Based on the
X
Y
B
D
Similarity statement and the
102
12
9
Prosperities of parallelograms,
E
F
We know the corresponding
78
W
Z
Angles are congruent.
So we can conclude:
X  Z  F  102
Y  E  D  78
To find the missing sides, we can use the properties of parallelograms and the
similarity theorems.
XW XY

BE BD
Corresponding Sides are proportional in similar
polygons
12 8

9 x
Place in the length of each side into the equation.
12 x  72
Cross Product Property
x6
Multiplication property of Equality.
We can now conclude the lengths of the following sides:
BD  EF  6
WZ  8
YZ  12
DF  9
12
3
The Similarity Ratio is
after reducing the fraction. (This can also be
or
9
2
written as 3:2)
Example 3
Determine if the triangles are similar. If the triangles are similar, what is the
similarity ratio?
Answer: If the triangles are similar, then
M
T
following would be true.
7
N
8
12
R
Y
13
9.33
0.857=0.857=0.923 all values are not
Equal, so the triangles are not similar.
6
P
RP RM MP


TY
YE
ET
6
8
12
Fill in the value of length.


7 9.33 13
E
N
A
Example 4 If MCB HAT , we know AT  10 , MC  16 , and CB  14 . What is the
length of HA ?
Answer Based on the statement above and that the corresponding angles and
sides must go in the correct order, meaning that M  H , C  A, and B  T .
Also we know that the following corresponding sides are proportional.
MC CB MB


HA AT HT
The circled proportion will be used to help us solve the
problem.
16 14

HA 10
Set up the proportion.
160  14( HA)
Cross Product Property
Divide both sides by 14. Multiplication property of
equality.
11.43  HA
Worksheet problems are on the next page
II. Practice Problems
For the figures below, write the similarity statement and determine the
conclusions regarding corresponding sides and angles.
E
A
1.
2.
G
F
A
H
E
D
M
C
B
P
T
L
N
3.
4.
MATH
RULZ
Draw your own figure.
L
P
E
F
Directions: Determine if the following polygons are similar. If they are similar,
write the appropriate similarity statement. What is the similarity ratio?
30
Y
Z
B
5.
6.
14
G
12
H
F
18
24
J
10
I
H
7
24
I
A
C
52
40
5
K
D
9
J
X
W
E
7. Rectangles ABCD and WXYZ
AD  30, AB  64,WZ  50,WX  96
8. JMR and KNP
J and N are right angles
R  67, P  23, MJ  24, MR  26, RJ  10
KN  15, NP  36, and KP  39
Directions : Determine whether the statement is true or false.
9. Two right triangles are similar.
10. Two squares are similar.
11. If two polygons are congruent, then the polygons are similar.
12. If two polygons are similar, then the two polygons are congruent.
Find the value of x in the figures below.
13.
3x-8
6
15
2x-3
14.
ABCD
EFGH
AB  4, BC  3, EF  x  3, FG  2 x  4
III. Challenge Problems
15. A ballroom is 100 feet long by 75 feet wide. On a blueprint for the ballroom,
the room is 7 inches long. What is the width of the ballroom on the blueprint if it
similar to the actual ballroom
16. The dimensions of the United States is 13937 kilometers in width and 2450
kilometers in length. You teacher wants you to draw a scale diagram on a 8 ½
inch by 11 inch piece of paper. She suggested that you use a scale of 1 inch
equals 1200 kilometers. Will the map fit on the paper? Explain why or why not.
If it does not fit, what scale should you use?
IV. Answers
Quad DACB
Quad FGHE;
Corresponding Angles:D  F ; A  G; C  H ; B  E
Corresponding Sides :
AMP
DA AC CB BD



FG GH HE EF
ELT ;
2. Corresponding Angles:A  E; M  L; P  T
Corresponding Sides :
NIP
AM MP AP


EL
LT ET
NEF ;
3. Corresponding Angles:N  N ; I  E; P  F
Corresponding Sides :
Quad MATH
NI
IP NP


NE EF NF
Quad RULZ ;
4. Corresponding Angles:M  R; A  U ; T  L; H  Z
Corresponding Sides :
MA AT TH MH



RU UL LZ
RZ
5. The polygons are similar. They have a similarity ratio of 2:1. FGHIJ BAEDH
6. The polygons are not similar.
7. The quadrilaterals are not similar.
8.The triangles are similar. The similarity ratio is 3:2 JMR NPK
9. False
13. x   1 4
10. True
11. True
12. False.
14. x  5
15. The width on the blueprint of the ballroom will be 5.25 inches.
16. The map would not fit on the paper. The length would be greater than 11
inches long when scaled down. Increasing the scale to 1 inch to 1300 kilometers
would make the map fit on the 8 ½ by 11 inch piece of paper.