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Chapter 8
Similarity
Chapter 8 Objectives
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Define a ratio
Manipulate proportions
Use proportions to solve geometric situations
Calculate geometric mean
Identify similar polygons
Prove triangles are similar
Use properties of similar triangles
Perform dilations
Lesson 8.1
Ratio
and
Proportion
Ratio
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If a and b are two quantities measured in the
same units, then the
ratio of a to b is a/b.
–
It can also be written as a:b.
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A ratio is a fraction, so the denominator cannot be zero.
Ratios should always be written in simplified
form.
–
5/
1/

10
2
Simplifying Ratios
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Not only should ratios be in simplified form, but they
must also be in the same units!
Example 1
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12 cm/
4m
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–
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Make sure they units are the same before simplifying the
numbers!
12 cm/
4 m(100 cm)
= 12 cm/400 cm  3 cm/100 cm
Some info to keep in mind when changing units
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100 cm = 1 m
1000 m = 1 km
12 in = 1 ft
3 ft = 1 yd
5280 ft = 1 mile
16 oz = 1 lb
Example 2
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Sometimes you may be given a problem that states the ratios of
side lengths or angle measures.
The ratio of the measures of the angles in a triangle
are 1:2:3. Find the measures of all three angles.
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You must set one of the angles equal to x and adjust the other
according to the ratio.
60o
2x
3x
90o
x
x + 2x + 3x = 180o
6x = 180o
x = 30
Proportion
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An equation that has two ratios equal to each
other is called a proportion.
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A proportion can be broken down into two parts.
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Extremes
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Which is the numerator of the first ratio and the denominator
of the second ratio
Means
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Which is the denominator of the first and numerator of the
second.
a
c
=
b
d
Properties Of Proportions
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Cross Product Property
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The product of the
extremes equals the
product of the means.
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Also known as crossmultiplying.
a
c
=
b
d
ad = bc
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Reciprocal Property
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Taking the reciprocal of
the entire proportion
creates an equivalent
proportion.
a
cd
b
=
b
d
a
c
Solving Proportions
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To solve a proportion, you must use the cross product property.
So multiply the extremes together and set them equal to the
means.
a
c
=
b
d
ad = bc
Example 3
Solve the following proportions using the Cross
Product Property
I.
2
x
II.
2
b+3
=
=
3
9
4
b
Lesson 8.3
Similar Polygons
Similarity of Polygons
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Two polygons are similar when the following
two conditions exist
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Corresponding angles are congruent
Correspondng sides are proportional
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Means that all side fit the same ratio.
The symbol for similarity is
– ~
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ABCD ~ EFGH
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This is called a similarity statement.
Proportional Statements
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Proportional statements are written by identifying all
ratios of corresponding sides of the polygons.
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Assume that square ABCD  EFGH
AB/
EF
A
B
D
C
= BC/FG = CD/GH = AD/EH
E
F
H
G
Scale Factor
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Since all the ratios should be equivalent to each other,
they form what is called the scale factor.
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•
We represent scale factor with the letter k.
This is most easily found by find the ratio of one pair
of corresponding side lengths.
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Be sure you know the polygons are similar.
A
B
5
5
D
6
C
E
F
k = 20/5
k=4
20
20
24
H
G
Theorem 8.1:
Similar Perimeters
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If two polygons are similar, then the ratio of
their perimeters is equal to the ratio of their
side lengths.
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This means that if you can find the ratio of one pair
of corresponding sides, that is the same ratio for
the perimeters.
Lesson 8.4
Similar Triangles
Postulate 25:
Angle-Angle Similarity Postulate
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If two angles of one triangle are congruent to
two angles of another triangle, then the two
triangles are similar.
Example 4
Find the length of side RT
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18
N
P
>
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12
Q
6
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8
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>
R
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x/
18
6/
12
=
12x = (18)(6)
12x = 108
x=9
Be sure that NPQ  TRQ.
To do so try to use AA to
find two angles that are
congruent to each other.
T
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Hint: Parallel Line Postulates
RQT  PQN
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Vertical Angles
QNP  QTR
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Alternate Interior Angles
NPQ  TRQ
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AA Similarity
Lesson 8.5
Proving Triangles are Similar
Theorem 8.2:
Side-Side-Side Similarity
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If the corresponding sides of two triangles are
proportional, then the triangles are similar.
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Your job is to verify that all corresponding sides fit the same
exact ratio!
10
10
6
5
5
3
Theorem 8.3:
Side-Angle-Side Similarity
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If an angle of one triangle is congruent to an angle of a
second triangle and the lengths of the sides including
these angles are proportional, then the triangles are
similar.
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Your task is to verify that two sides fit the same exact ratio
and the angles between those two sides are congruent!
10
6
5
3
Example 5
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Identify the similar triangles, if any. If so, explain how
you know they are similar and write a similarity
statement.
G
B
6
8
5
12
I
A
E
4
3.75
C
F
6
3
ABC  DEF, by SSS Similarity
D
8
H
Using Theorems
8.2 and 8.3
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These theorems share the abbreviations with those
from proving triangles congruent in chapter 4.
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So you now must be more specific
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SSS
SAS
SSS Congruence
SSS Similarity
SAS Congruence
SAS Similarity
You chose based on what are you trying to show?
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Congruence
Similarity
Lesson 8.6
Proportions
and
Similar Triangles
Triangle Proportionality
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Theorem 8.4:
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Converse of Triangle Proportionality
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If a line divides two sides
proportionally, then it is parallel to
the third side.
Triangle Proportionality
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Theorem 8.5:
If a line parallel to one
side of a triangle
intersects the other two
sides, then it divides the
two sides proportionally.
If RT/TQ = RU/US,
then TU // QS.
R
If TU // QS, then
RT/
RU/ .
=
TQ
US
T
Q
U
S
Example 6
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Determine what they are asking for
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If they are asking to solve for x
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If they are asking if the sides are parallel
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x
10/2
Make sure you know the sides are parallel!
Make sure you know the ratio of sides lengths are the
same.
R
10/
x4
=
4x = (10)(2)
4x==2020
4x
xx
= 5= 5
/4 = /2
x
T
Q
2
10
U
4
S
Theorem 8.6:
Proportional Transversals
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If three parallel lines intersect two
transversals, then they divide the transversals
proportionally.
Lesson 8.7
Dilations
Dilation
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A dilation is a transformation with the following
properties
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If point P is not at the center C, then the image P’ lies on
ray CP.
If point P is at the center, then P = P’.
A dilation is something that will increase or
decrease the size of the figure while still
maintaining similarity.
P’
P
C
Scale Factor of a Dilation
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The scale factor of a dilation is found by the
following
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k = CP’/CP
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k stands for scale factor
It is basically the distance from the center to the
image divided by the distance from the center to
the pre-image.
12
12/
k=
k=4
3
C
3
P
P’
Reduction or Enlargement
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A reduction is when the image
is smaller than the pre-image.
The scale factor will be a
number between 0 and 1.
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0<k<1
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An enlargement is when
the image is larger that
the pre-image.
The scale factor will be a
number greater than 1.
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C
k>1
C
Scale Factor with Coordinates
Center at the Origin
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When applying the scale factor to a set of
coordinates, simply distribute to both the x and
y values of each coordinate.
Example
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Perform the following dilation for point P
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k = 3, P(2,7)
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P’(3•2,3•7)
» P’(6,21)