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Geometry
8.3 Similar Polygons
Goals



Identify similar polygons
Find the ratio of similarity between similar
figures.
Solve problems involving similar figures.
May 23, 2017
Geometry 8.3 Similar Polygons
2
Similar Polygons



ABCD and RSTV are similar polygons:
Corresponding angles are congruent.
Corresponding sides are proportional.
9
A
B
6
R
9
C
15
D
6
T
10
8
V
12
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S
Geometry 8.3 Similar Polygons
3
Similar Polygons


Corresponding angles are congruent:
A  R, B  S, C  T, D  V
9
A
B
6
R
9
C
15
D
6
T
10
8
V
12
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S
Geometry 8.3 Similar Polygons
4
Similar Polygons

Corresponding sides
are proportional:
9
A
15 12 9 9

 
10 8 6 6
3 3 3 3
  
2 2 2 2
B
6
R
9
C
15
D
6
T
10
8
V
12
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S
Geometry 8.3 Similar Polygons
5
Similar Polygons



9
A
Corr. s 
Sides prop.
ABCD ~ RSTV
B
6
R
9
C
15
D
6
T
10
8
V
12
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S
Geometry 8.3 Similar Polygons
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Similar Polygons




Corresponding Angles are congruent.
Corresponding Sides are proportional.
Use the symbol “~” for similar.
To show that two polygons are similar, you
must prove both things: angles congruent,
sides proportional.
May 23, 2017
Geometry 8.3 Similar Polygons
7
Similarity Statements


List the congruent
J  Q, K  S, L  R
angles.
Write the ratios of the
JK JL KL


corresponding sides.
QS QR SR
K
S
70
70
J
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L
Q
Geometry 8.3 Similar Polygons
R
8
Example





Are these figures similar?
Yes
Why?
Corr. angles congruent
Corr. sides proportional.
2
H
N
Geometry 8.3 Similar Polygons
F
1.5
4
6
4
O
May 23, 2017
3
E
G
M
3
8
P
9
Write the similarity statements.
E 3
E  N, F  M
2
G  P, H  O
H
4
EF
FG GH HE N
6
=


NM MP PO ON 4
EFGH ~ NMPO
O
8
F
1.5
G
M
3
P
Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc.
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Geometry 8.3 Similar Polygons
10
Scale Factor
If two polygons are
similar, the ratio of two
corresponding sides is
called the scale factor.
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Geometry 8.3 Similar Polygons
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Scale Factor


The scale factor of JKL to QSR is
10/5 or 2/1.
The scale factor of QSR to JKL is
5/10 or 1/2.
K
10
J
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S
70
5 70
L
Q
Geometry 8.3 Similar Polygons
R
12
Perimeter and Similar Figures
1. ABCD ~ FGHI
2. Find the scale factor of ABCD to FGHI.
3. Find the values of x, y, and z.
4. Find the perimeter of ABCD and FGHI.
5. Find the ratio of the perimeters.
A
10
B
F 5 G
14
14
x
I z H
D 4 C
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y
Geometry 8.3 Similar Polygons
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Perimeter and Similar Figures
2. Find the scale factor from ABCD to FGHI.
10 2

5 1
The only known corresponding
sides are AB and FG.
A
10
B
F 5 G
14
14
x
I z H
D 4 C
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y
Geometry 8.3 Similar Polygons
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Perimeter and Similar Figures
3. Find the values of x, y, and z.
2 14

1 y
2 y  14
y7
10 14

5
x
10 x  70
x7
A
10
2 4

1 z
2z  4
z2
B
F 5 G
14
14
x
I z H
D 4 C
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y
Geometry 8.3 Similar Polygons
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Perimeter and Similar Figures
4. Find the perimeter of ABCD and FGHI.
P = 42
A
10
B
P = 21
F 5 G
14
14
7
I 2 H
D 4 C
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Geometry 8.3 Similar Polygons
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Perimeter and Similar Figures
5. Find the ratio of the perimeters.
Ratio of perimeters 2:1
Ratio of Similarity
P = 42
A
10
B
2:1
P = 21
F 5 G
14
14
7
I 2 H
D 4 C
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Geometry 8.3 Similar Polygons
17
Theorem 8.1

If two polygons are similar, then the ratio of
their perimeters is equal to the ratios of their
corresponding side lengths and is equal to
the similarity ratio.
May 23, 2017
Geometry 8.3 Similar Polygons
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Theorem 8.1 Example
20
These figures are similar.
Find the perimeter of the
smaller one.
8
P = 100
May 23, 2017
P=?
Geometry 8.3 Similar Polygons
20 100

8
P
20 P  800
P  40
19
Problems to Solve
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Geometry 8.3 Similar Polygons
20
Problem 1
Solve for x and y if the triangles are similar.
x+6
20
4
y–2
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8
6
Geometry 8.3 Similar Polygons
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Problem 1 Solution
Scale Factor is 20/8
x+6
20
8
4
y–2
20Solve
x6

8 for x4
8( x  6)  80
x  6  10
x4
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6
20 Solve
y2

8 for y6
8( y  2)  120
y  2  15
y  17
Geometry 8.3 Similar Polygons
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Problem 2
Find x and y if the figures are similar.
x + 10
85
100
32
60
24
95
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Geometry 8.3 Similar Polygons
y
23
Problem 2 Solution
x + 10
85
100
60
24
95
Similarity Ratio
y = 360 - 100 - 85 - 95
y = 80
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Geometry 8.3 Similar Polygons
y
60 x  10

24
32
1920  24 x  240
1680  24 x
x  70
24
Problem 3





ABC ~ RST
AB = 20
ST = 4
BC = RS
Find BC and RS.
May 23, 2017
Geometry 8.3 Similar Polygons
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Problem 3 Solution





ABC ~ RST
AB = 20
ST = 4
BC = RS
Find BC and RS.
May 23, 2017
A
R
20
x
B
x
Geometry 8.3 Similar Polygons
C S
4
T
26
Problem 3 Solution
A
AB BC

RS ST
20 x

x 4
x 2  80
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R
20
x
x  80  16  5
B
x4 5
x  8.94
ABC ~ RST
x
Geometry 8.3 Similar Polygons
C S
4
T
27
Problem 4
You want to print a picture in your camera. You
have two sizes of paper for your printer: 4 × 6
and 5 × 7. Does it matter? Will the pictures
printed from each size of paper be similar?
4×6
5×7
May 23, 2017
4 6

5 7
28  30
Geometry 8.3 Similar Polygons
Sides not
proportional,
figures not
similar.
28
Problem 5
MNOP has a perimeter of 24. Find the perimeter
of QRST if MN = 8 and QR = 12.
MN 8

QR 12
8 24

12 P
8 P  288
P  36
May 23, 2017
Geometry 8.3 Similar Polygons
29
Summary




Two polygons are similar if they have the
same shape, but a different size.
If polygons are similar corresponding angles
are congruent, and corresponding sides are
proportional.
The ratio of any two corresponding sides is
the scale factor.
The ratio of the perimeters is equal to the
ratio of two corresponding sides.
May 23, 2017
Geometry 8.3 Similar Polygons
30