The student is able to identify scale factors and use them to solve problems involving similar polygons. Scale factors allow comparison of corresponding side lengths of similar polygons through proportions. To determine if two polygons are similar, their corresponding angles must be congruent and side lengths must be proportional as shown through equal ratios.

1. Similar Polygons
The student is able to (I can):
• Identify scale factors and use scale factors to solve
problems
• Write and simplify ratios
• Use proportions to solve problems
• Identify similar polygons
• Use similarity to solve problems.

2. ratioratioratioratio – a comparison of two numbers by division.
The ratio of two numbers a and b, where b does not equal 0
(b ≠ 0) can be written as
a to b
a : b
Example: The ratio comparing 1 and 2 can be written 1 to 2,
1 : 2, or .
To compare more than two numbers, use “dot”
notation. Ex. 3 : 7 : 9
a
b
1
2

3. proportionproportionproportionproportion – an equation stating that two ratios are equal.
Two sets of numbers are proportionalproportionalproportionalproportional if they use the same
ratio.
Example: or a : b = c : d
Cross Products Property
In a proportion, if , and b and d ≠ 0, then ad = bc
scale factorscale factorscale factorscale factor – the ratio of the image to the preimage (k)
If k < 1, the figure gets smaller; if k > 1, the figure gets larger.

a c
b d
a c
b d


4. Examples
1. What is the scale factor between the two rectangles?
2. If you are enlarging a 4x6 photo by a scale factor of 4,
what are the new dimensions?
10
24
5
12

5. Examples
1. What is the scale factor between the two rectangles?
2. If you are enlarging a 4x6 photo by a scale factor of 4,
what are the new dimensions?
4(4) = 16 6(4) = 24
New dimensions = 16x24
10
24
5
12
5 1 12 1
(or )
10 2 24 2
k k   

6. Examples Solve each proportion:
1.
2.
3.
3
8 32
x

4 2
5x

2
6 3
x x 


7. Examples Solve each proportion:
1.
8x = 96 x = 12
2.
2x = 20 x = 10
3.
3x = 6(x – 2)
3x = 6x – 12
–3x = –12 x = 4
3
8 32
x

4 2
5x

2
6 3
x x 


8. Examples 4. The ratio of the angles of a triangle is
2: 2: 5. What is the measure of each
angle?

9. Examples 4. The ratio of the angles of a triangle is
2: 2: 5. What is the measure of each
angle?
2x + 2x + 5x = 180˚
9x = 180˚
x = 20
220 = 40˚
220 = 40˚
520 = 100˚

10. similarsimilarsimilarsimilar polygonspolygonspolygonspolygons – two polygons are similar if and only if their
corresponding angles are congruent and their
corresponding side lengths are proportional.
Example: Show that the two polygons are similar.
S  B
P  R
N  A
G  K
S P
NG
B R
AK
6
5
3
4
12
10
6
8
3 4 5 6
6 8 10 12
  
SPNG ~ BRAK

11. A similarity statementsimilarity statementsimilarity statementsimilarity statement describes two similar polygons by
listing their corresponding vertices.
Example: SPNG ~ BRAK
To check whether two ratios are equal, cross-multiply
them—the products should be equal.
Example:

3 4
?
6 8
24 24 

12. Example
Determine whether the rectangles are similar. If so, write
the similarity ratio and a similarity statement.
All of the angles are right angles, so all the angles are
congruent.
150 = 150 sim. ratio: QUAD ~ RECT
Q
U
A
D
R E
CT
15
6
25
10
6 15
?
10 25

3
5