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Chapter 8
Similarity
8.1
Ratio and Proportion
Ratios
Ratio- Comparison of 2 quantities in the
same units
 The ratio of a to b can be written as

a/b
 a : b


The denominator cannot be zero
Simplifying Ratios

Ratios should be expressed in simplified form


6:8 = 3:4
Before reducing, make sure that the units are
the same.

12in : 3 ft
12in : 36 in
1: 3
Examples (page 461)

Simplify each ratio
10.
16 students
24 students
12.
22 feet
52 feet
18.
60 cm
1m
Examples (page 461)

Simplify each ratio
20.
2 mi
3000 ft
24.
20 oz.
4 lb
There are 5280 ft in 1 mi.
There are 16 oz in 1 lb.
Examples (page 461)

Find the width to length ratio
14.
16 mm
20 mm
16.
12 in.
2 ft
Using Ratios Example 1

The perimeter of the isosceles triangle
shown is 56 in. The ratio of LM : MN is
5:4. Find the length of the sides and
the base of the triangle.
L
N
M
Using Ratios Example 2

The measures of the angles in a
triangle are in the extended ratio 3:4:8.
Find the measures of the angles
4x
8x
3x
Using Ratios Example 3

The ratios of the side lengths of ΔQRS
to the corresponding side lengths of
ΔVTU are 3:2. Find the unknown
lengths.
Q
U
T
2 cm
V
S
18 cm
R
Proportions

Proportion
Ratio = Ratio
 Fraction = Fraction

Means and Extremes

Extreme: Mean = Mean: Extreme
Extreme
Mean

Mean
Extreme
Solving Proportions

Solving Proportions



Cross multiply
Let the means equal the extremes
Example:
3
5

x
20
Properties of Proportions

Cross Product Property
a c
If  , then ad  bc
b d

Reciprocal Property
a c
b d
If  , then 
b d
a c
Solving Proportions Example 1
9 6

14 x
Solving Proportions Example 2
s5 s

4
10
Solving Proportions Example 3
A photo of a building
has the
measurements
shown. The actual
building is 26 ¼ ft
wide. How tall is
it?
2.75 in
1 7/8 in
8.2
Problem solving in
Geometry with Proportions
Properties of Proportions
a c
a b
If  , then 
b d
c d
a c
ab cd
If  , then

b d
b
d
Example 1

Tell whether the statement is true or
false

A.
s 15
s 3
If
 , then 
10 t
t 2

B.
3 5
3 x 5 y
If  , then

x y
x
y
Example 2

In the diagram MQ LQ

MN LP
Find the length of LQ.
M
6
N
15
13
Q
L
5 P
Geometric Mean

Geometric Mean

The geometric mean between two
numbers a and b is the positive number x
such that
a x

x b
ex: 8/4 = 4/2
Example 3

Find the geometric mean between 4
and 9.
Similar Polygons
Polygons are similar if and only if
the corresponding angles are congruent
and
 the corresponding sides are
proportionate.



Similar figures are dilations of each
other. (They are reduced or enlarged
by a scale factor.)
The symbol for similar is 
Example 1
Determine if the sides of the polygon are
proportionate.
8m
12 m
6m
8m
6m
Example 2
Determine if the sides of the polygon
are proportionate.
15 m
9m
5m
3m
12 m
4m
Example 3
Find the missing measurements.
HAPIE  NWYRS
6 A
H
18
P
5
AP =
EI =
SN =
YR =
E
I
4
W
24
Y
N
S
21
R
Example 4
Find the missing measurements.
QUAD  SIML
A
D
S 12
L
65º
8
125º
Q 20
25
I 95º
U
M
QD =
MI =
mD =
mU =
mA =
8.4/8.5
Similar Triangles
Similar Triangles

To be similar, corresponding sides
must be proportional and
corresponding angles are congruent.
Similarity Shortcuts
AA Similarity Shortcut
If two angles in one triangle are
congruent to two angles in another
triangle, then the triangles are similar.
Similarity Shortcuts
SSS Similarity Shortcut
If three sides in one triangle are
proportional to the three sides in
another triangle, then the triangles are
similar.
Similarity Shortcuts
SAS Similarity Shortcut
If two sides of one triangle are
proportional to two sides of another
triangle and
their included angles are congruent,
then the triangles are similar.
Similarity Shortcuts
We have three shortcuts:
AA
SAS
SSS
Example 1
9
g
6
4
7
10.5
Example 2
k
32
h
50
24
30
Example 3
36
42
m
24
4. A flagpole 4 meters tall casts a 6 meter
shadow. At the same time of day, a nearby
building casts a 24 meter shadow. How tall is
the building?
4
m
6m
24m
5. Five foot tall Melody casts an 84 inch
shadow. How tall is her friend if, at the same
time of day, his shadow is 1 foot shorter than
hers?
6. A 10 meter rope from the top of a flagpole
reaches to the end of the flagpole’s 6 meter
shadow. How tall is the nearby football
goalpost if, at the same moment, it has a
shadow of 4 meters?
10m
6m
4m
7. Private eye Samantha Diamond places a
mirror on the ground between herself and an
apartment building and stands so that when
she looks into the mirror, she sees into a
window. The mirror is 1.22 meters from her
feet and 7.32 meters from the base of the
building. Sam’s eye is 1.82 meters above the
ground. How high is the window?
1.82
1.22
7.32
8.6
Proportions and Similar
Triangles
Proportions

Using similar triangles missing sides
can be found by setting up proportions.
Theorem

Triangle Proportionality Theorem

If a line parallel to one side of a triangle
intersects the other two sides, then it
divides the two sides proportionally.
Q
T
RT RU
If TU || QS ,then

.
TQ US
R
S
U
Theorem

Converse of the Triangle Proportionality
Theorem

Q
If a line divides two sides of a triangle
proportionally, then it is parallel to the
third side.
RT RU
T
If
TQ
R
S
U

US
, thenTU || QS .
Example 1

In the diagram, segment UY is parallel
to segment VX, UV = 3, UW = 18 and
XW = 16. What is the length of
segment YX?
U
V
W
Y
X
Example 2

Given the diagram, determine whether
segment PQ is parallel to segment TR.
Q
9.75
P
R
9
26
T
24
S
Theorem

If three parallel lines intersect two
transversals, then they divide the
transversals proportionally.
Theorem

If a ray bisects an angle of a triangle,
then it divides the opposite side into
segments whose lengths are
proportional to the lengths of the other
two sides.
Example 3

In the diagram, 1  2  3, AB =6,
BC=9, EF=8. What is x?
C
9
B
6
A
1
D
3
2
8
x
E
F
Example 4

In the diagram, LKM  MKN. Use
the given side lengths to find the length
of segment MN.
15
L
N
M
3
17
K
5. Juanita, who is 1.82 meters tall, wants
to find the height of a tree in her
backyard. From the tree’s base, she
walks 12.20 meters along the tree’s
shadow to a position where the end of
her shadow exactly overlaps the end of
the tree’s shadow. She is now 6.10
meters from the end of the shadows.
How tall is the tree?
6.10
1.82
12.20
8.7
Dilations
C
3
P
9
P’
Dilations
Dilation: Transformation that maps all
points so that the proportion CP ' stands
CP
true.
 Enlargement: A dilation which makes
the transformed image larger than the
original image
 Reduction: A dilation which makes the
transformed image smaller than the
original image.

Enlargement
An enlargement has a scale factor of k which if found by the
proportion CP
. ' In an enlargement k is always greater than 1.
CP
Find k:
C
3
P
9
P’
Reduction

A reduction has a scale factor of k
CP '
which is found by the proportion
. In
CP
a reduction, 0 < k < 1.
C
P
Find k
6
P’
14
Dilations in a coordinate plane
If the center of the dilation is the origin,
the image can be found by multiplying
each coordinate by the scale factor
Example:
Original coordinates:
(3, 6), (6, 12) and (9, 3)
Scale factor: 1/3
Find the image coordinates.
