Geometry
8.3 Similar Polygons
Goals



Identify similar polygons
Find the ratio of similarity between similar
figures.
Solve problems involving similar figures.
March 23, 2016
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
March 23, 2016
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
March 23, 2016
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
March 23, 2016
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
March 23, 2016
S
Geometry 8.3 Similar Polygons
6
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.
March 23, 2016
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
March 23, 2016
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
March 23, 2016
3
E
G
M
3
8
P
9
Write the similarity statements.
E  N, F  M
2
G  P, H  O
EF
FG GH HE
=


NM MP PO ON
EFGH ~ NMPO
3
E
H
N
1.5
4
6
4
O
F
G
M
3
8
P
Or: HEFG ~ ONMP, GFEH ~ PMNO, EHGF ~ NOPM, etc.
March 23, 2016
Geometry 8.3 Similar Polygons
10
Scale Factor
If two polygons are
similar, the ratio of two
corresponding sides is
called the scale factor.
March 23, 2016
Geometry 8.3 Similar Polygons
11
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
March 23, 2016
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
March 23, 2016
y
Geometry 8.3 Similar Polygons
13
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
March 23, 2016
y
Geometry 8.3 Similar Polygons
14
Perimeter and Similar Figures
3. Find the values of x, y, and z.
10 14

5
x
10 x  70
2 14

1 y
2 y  14
x7
y7
A
10
2 4

1 z
2z  4
z2
B
F 5 G
14
14
x
I z H
D 4 C
March 23, 2016
y
Geometry 8.3 Similar Polygons
15
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
March 23, 2016
7
Geometry 8.3 Similar Polygons
16
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
March 23, 2016
7
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.
March 23, 2016
Geometry 8.3 Similar Polygons
18
Theorem 8.1 Example
20
These figures are similar.
Find the perimeter of the
smaller one.
8
P = 100
P=?
20 100

8
P
20 P  800
P  40
March 23, 2016
Geometry 8.3 Similar Polygons
19
Problems to Solve
March 23, 2016
Geometry 8.3 Similar Polygons
20
Problem 1
Solve for x and y if the triangles are similar.
x+6
20
4
y–2
March 23, 2016
8
6
Geometry 8.3 Similar Polygons
21
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
March 23, 2016
6
20 Solve
y2

8 for y6
8( y  2)  120
y  2  15
y  17
Geometry 8.3 Similar Polygons
22
Problem 2
Find x and y if the figures are similar.
x + 10
85
100
32
60
24
95
March 23, 2016
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
March 23, 2016
32
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.
March 23, 2016
Geometry 8.3 Similar Polygons
25
Problem 3 Solution





ABC ~ RST
AB = 20
ST = 4
BC = RS
Find BC and RS.
March 23, 2016
A
R
20
x
B
x
Geometry 8.3 Similar Polygons
C S
4
T
26
Problem 3 Solution
AB BC

RS ST
20 x

x 4
x 2  80
A
R
20
x
x  80  16  5
B
x4 5
ABC ~ RST
x
C S
4
T
x  8.94
March 23, 2016
Geometry 8.3 Similar Polygons
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
March 23, 2016
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
March 23, 2016
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.
March 23, 2016
Geometry 8.3 Similar Polygons
30