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Chapter 8
Similarity
Chapter 8 Objectives
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
Lesson 8.1 Objectives
Define a ratio
Derive proportions from ratios
Solve proportions
Ratio
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.

A ratio is a fraction, so the denominator cannot
be zero.
Ratios should always be written in
simplified form.
 5/10  1/2
Simplifying Ratios
Not only should ratios be in simplified form, but they
must also be in the same units!
 12 cm/4 m

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






100 cm = 1 m
1000 m = 1 km
12 in = 1 ft
3 ft = 1 yd
5280 ft = 1 mile
16 oz = 1 lb
Using Ratios
Sometimes you may be given a problem that states
the ratios of side lengths or angle measures.

Example:

The ratio of the measures of the angles in a triangle are 1:2:3.
Find the measures of all three angles.

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
An equation that has two ratios equal to each
other is called a proportion.

A proportion can be broken down into two parts.

Extremes


Which is the top (numerator) of the first ratio and the
bottom (denominator) of the second ratio
Means

Which is the denominator of the first and numerator of the
second.
a
c
=
b
d
Properties Of Proportions
Cross Product
Property

The product of the
extremes equals the
product of the means.

Reciprocal Property

Taking the reciprocal of
the entire proportion
creates an equivalent
proportion.
Also known as crossmultiplying.
a
c
=
b
d
ad = bc
a
cd
b
=
b
d
a
c
Solving Proportions
To solve a proportion, you must use the cross product
property (or cross multiply).

So multiply the extremes together and set them equal to the
means.
a
c
=
b
d
ad = bc
Homework 8.1
1-44
skip 2,3
 p461-462

In Class – 8,17,25,43
Due Tomorrow
Lesson 8.2
Problem Solving
in Geometry
with Proportions
Lesson 8.2 Objectives
Utilize properties of proportions
Calculate geometric mean
Model real-life proportions
Additional Properties of Proportions
Means Exchange

The means of a
proportion can swap
positions and still
maintain an
equivalent
proportion.
a
cb
=
c
b
d
Numerator Adding
Denominator

Adding the
appropriate
denominator to its
corresponding
numerator maintains
an equivalent
proportion.
a
c+d
a+b
c
=
b
d
Geometric Mean
The geometric mean of two positive numbers a and
b is the positive number x such that:
geometric “mean(s)”
To solve,
you still
cross
multiply.
a
x
=
b
x
x2 = (a)(b)
x =  (a)(b)
Using Proportions
Example 4

p467
A scale model of the Titanic is 107.5 inches long and 11.25
inches wide. The Titanic itself was 882.75 feet long. How wide
was it?
Every ratio in the proportion should be set up so that they follow the
same pattern.
Length of Titanic
Width of Titanic
=
Width of Model
You determine
the pattern!
x
11.25 in
=
Length of Model
882.75 ft
107.5 in
(107.5 in) x = (11.25 in)(882.75 ft)
107.5 in
x = 92.4 ft
107.5 in
Inches cancel
out just like
numbers do!
Another Example of Proportions
L
Problem 27

M
12
16
J
K
p469
P
GIVEN: LJ/JN = MK/KP, find JN. N
Follow the proportion given to you.
Then plug in the numbers appropriately
LJ
JN
9(16-JN) = 12(JN)
144 - 9(JN) = 12(JN)
+9(JN)
+9(JN)
144 = 21(JN)
21
JN =
21
48/
7
9
MK
=
16 - JN
JN
KP
12
=
9
Homework 8.2
1-30

p468-469
In Class – 9,13,17,27
Due Tomorrow

Double HW Check Tomorrow

8.1-8.2
Lesson 8.3
Similar Polygons
Lesson 8.3 Objectives
Identify similar polygons
Calculate scale factor
Similarity of Polygons
Two polygons are similar when the following
two conditions exist


Corresponding angles are congruent
Correspondng sides are proportional

Means that all side fit the same ratio.
The symbol for similarity is
 ~

ABCD ~ EFGH

This is called a similarity statement.
Proportional Statements
Proportional statements are written by
identifying all ratios of corresponding sides of
the polygons.
AB/
EF
A
B
D
C
= BC/FG = CD/GH = AD/EH
E
F
H
G
Scale Factor
Since all the ratios should be equivalent to each
other, they form what is called the scale factor.

We represent scale factor with the letter k.
This is most easily found by find the ratio of one pair
of corresponding side lengths.
A

Be sure you know the polygons are similar.
B
E
F
5
D
6
C
k = 20/5
k=4
20
24
H
G
Homework 8.3a
2-18

p475-476
In Class – 5,9,11
Due Tomorrow
Quiz Wednesday

Lessons 81.-8.3
Theorem 8.1:
Similar Perimeters
If two polygons are similar, then the ratio of
their perimeters is equal to the ratio of their
side lengths.

This means that if you can find the ratio of one pair
of corresponding sides, that is the same ratio for
the perimeters.
Homework 8.3b
19-38

p476-477
In Class – 19,38
Due Tomorrow
Lesson 8.4
Similar Triangles
Lesson 8.4 Objectives
Identify similar triangles
Apply similar triangles to real world
situations
Postulate 25:
Angle-Angle Similarity Postulate
If two angles of one triangle are
congruent to two angles of another
triangle, then the two triangles are
similar.
Homework 8.4
1-17

skip 2,3,8

Was supposed to be done after the quiz!
18-28, 33-44

p483-485
In Class – 19,25,44
Due Tomorrow
Lesson 8.5
Proving Triangles are Similar
Lesson 8.5 Objectives
Prove triangles are similar using the
SSS Similarity Theorem
Prove Triangles are similar using the
SAS Similarity Theorem
Utilize properties of similar triangles
Theorem 8.2:
Side-Side-Side Similarity
If the corresponding sides of two triangles
are proportional, then the triangles are
similar.

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
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.

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
Using Theorems
8.2 and 8.3
These theorems share the abbreviations with those
from proving triangles congruent in chapter 4.


SSS
SAS
So you now must be more specific




SSS Congruence
SSS Similarity
SAS Congruence
SAS Similarity
You chose based on what are you trying to show?


Congruence
Similarity
Homework 8.5
1-14,19-25
skip 5
 p492-493

In Class – 3,7,10,15,19
Due Tomorrow
Quiz Wednesday

Lessons 8.4-8.6
Lesson 8.6
Proportions
and
Similar Triangles
Lesson 8.6 Objectives
Identify proportional components of
similar triangles
Use proportionality theorems to
calculate segment lengths
Triangle Proportionality
Theorem 8.4:
Theorem 8.5:
Triangle Proportionality
Converse of Triangle Proportionality
 If a line divides two sides
proportionally, then it is parallel to
the third side.

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
Using Theorems
8.4 and 8.5
Determine what they are asking for

If they are asking to solve for x


Make sure you know the sides are parallel!
If they are asking if the sides are parallel

Make sure you know the ratio of sides lengths are the
same.
R
10/
4
= x/2
4x = 20
x=5
x
T
Q
2
10
U
4
S
Theorem 8.6:
Proportional Transversals
If three parallel lines intersect two
transversals, then they divide the
transversals proportionally.
Homework 8.6
3-14, 21-28

p502-503
In Class – 3,11,15,24,25
Lesson 8.7
Dilations
Lesson 8.7 Objectives
Identify a dilation
Calculate the scale factor of a dilation
Dilation
A dilation is a transformation with the following
properties


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.
C
P
P’
Scale Factor of a Dilation
The scale factor of a dilation is found by the
following

k = CP’/CP

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=3
4
C
3
P
P’
Reduction or Enlargement
A reduction is when
the image is smaller
than the pre-image.
The scale factor will be
a number between 0
and 1.

C
0<k<1
An enlargement is
when the image is
larger that the preimage.
The scale factor will be
a number greater than
1.

k>1
C
Scale Factor with Coordinates
Center at the Origin
When applying the scale factor to a set
of coordinates, simply distribute to both
the x and y values of each coordinate.

k = 3, P(2,7)

P’(3•2,3•7) =

P’(6,21)
Homework 8.7
1-21

skip 2,3,6,7,16-19
WS 8.7
Both Due Tomorrow
Test Thursday

Chapter 8