Geometry
4.5 Dilations
4.5 Warm Up Day 1
Use the graph to find the indicated length.
1. Find the length of
2. Find the length of
𝐷𝐸 and 𝐸𝐹
𝐵𝐶 and 𝐴𝐶
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4.5 Dilations
4.5 Warm Up Day 2
Plot the points in a coordinate plane. Then determine
whether the quadrilaterals are congruent.
1. A(-3, 4), B(-3, 7), C(3, -4), D(3, -1)
2. E(7, -2), F(2, -2), G(3, -4), H(5, -4)
3. I(9, 2), J(0, 2), K(6, 9), L(6, 2)
4. M(7, -9), N(7, 0), P(8, -3), Q(-1, -3)
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4.5 Dilations
4.5 Essential Question
What does it mean to dilate a figure?
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4.5 Dilations
Goals


Identify Dilations
Make drawings using dilations.
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4.5 Dilations
Rigid Transformations

Rotations




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Translations
These were isometries:
The pre-image and the image were congruent.
4.5 Dilations
Dilation


Dilations are non-rigid transformations.
The pre-image and image are similar, but
not congruent.
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4.5 Dilations
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4.5 Dilations
Dilation
rgemen
Enla
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4.5 Dilations
t
Dilation
Reduction
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4.5 Dilations
Dilation Definition
A dilation with center C and scale factor k is a transformation that maps
every point P to a point P’ so that the following properties are true:
1. If P is not the center point C, then the image point P’ lies on CP. The
scale factor k is an integer such that
k  1 and
CP'
k=
CP
2. If P is the center point C, then P = P’.
3. The dilation is a reduction if 0 < |k| < 1, and an enlargement if |k| > 1.
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4.5 Dilations
Dilation
R
S
C
Center of Dilation
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T
4.5 Dilations
Dilation
2CR
R
CR
R
S
CR
C
Center of Dilation
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T
4.5 Dilations
Dilation
2CR
R
CR
CS
R
CR
2CS
S
CS
C
Center of Dilation
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T
4.5 Dilations
S
Dilation
2CR
R
CR
C
CT
Center of Dilation
CR
2CS
S
CS
T
2CT
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R
CS
CT
T
4.5 Dilations
S
Dilation
RST ~ RST
2CR
R
CR
C
CT
Center of Dilation
CR
2CS
S
CS
T
2CT
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R
CS
CT
T
4.5 Dilations
S
Enlargement
Dilation
2CR
R
CR
C
CT
Center of Dilation
R
CR
2CS
S
CS
T
2CT
CS
S
CT
T
CR ' CS ' CT ' 2


 =k Scale Factor
CR CS CT 1
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4.5 Dilations
Example 1
Reduction
What type of dilation is this?
G
F
F’
G’
C
K’
H’
H
K
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4.5 Dilations
F ' G ' 15 1
k


FG
45 3
F ' K ' 12 1
k


FK
36
3
G
Example 1
What is the scale factor?
45
F
F’
36
k<1
C
12
Reduction
K’
H’
H
K
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Notice:
15 G’
4.5 Dilations
Example 2
Find the scale factor of the dilation.
Then tell whether the dilation is a
reduction or an enlargement
𝐶𝑃′
𝑘=
𝐶𝑃
12
𝑘=
8
3
𝑘=
2
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4.5 Dilations
Notice:
k>1
Enlarge
ment
Your Turn
Find the scale factor of the dilation. Then
tell whether the dilation is a reduction or
an enlargement
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4.5 Dilations
𝐶𝑃′
𝑘=
𝐶𝑃 Notice:
18
𝑘=
k<1
30
3 Reduction
𝑘=
5
Remember:



November 24, 2015
The scale factor k is
Image Length
S. F. =
Pre − image Length
If 0 < |k| < 1 it’s a reduction.
If |k| > 1 it’s an enlargement.
4.5 Dilations
Coordinate Geometry

Use the origin (0, 0) as the center of dilation.
The image of P(x, y) is P’(kx, ky).
Notation: P(x, y)  P’(kx, ky).
Read: “P maps to P prime”

You need graph paper, a ruler, pencil.



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4.5 Dilations
Example 3
Graph ABC with
A(1, 1), B(3, 6), C(5, 4).
B
C
A
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4.5 Dilations
Example 3
B’
Using a scale factor of k = 2, locate
points A’, B’, and C’. P(x, y)  P’(kx, ky).
C’
𝑨(𝟏, 𝟏)  𝑨’(𝟐  𝟏, 𝟐  𝟏) = 𝑨’(𝟐, 𝟐)
𝑩(𝟑, 𝟔)  𝑩’(𝟐  𝟑, 𝟐  𝟔) = 𝑩’(𝟔, 𝟏𝟐)
𝑪(𝟓, 𝟒)  𝑪’(𝟐  𝟓, 𝟐  𝟒) = 𝑪’(𝟏𝟎, 𝟖)
B
C
A
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4.5 Dilations
A’
Example 3
B’
Draw ABC.
C’
B
C
A
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A’
4.5 Dilations
Example 3
B’
You’re done.
C’
Notice that rays drawn
from the center of
dilation (the origin)
through every preimage
point also passes through
the image point.
B
C
A
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A’
4.5 Dilations
Example 4
Graph quad. KLMN with K(−4, 8), L(0, 8), M(4, 4),
and N(−4, −4) and its image after a dilation with a
3
scale factor of
4
x, y →
3 3
x, y
4 4
K(−4, 8) →
K′(−3, 6)
L(0, 8) →
L′(0, 6)
M(4, 4) →
M′(3, 3)
N(−4, −4) →
N′(−3, −3)
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4.5 Dilations
Your Turn
T(0, 12)
Draw RSTV
with
R(0, 0)
S(6, 3)
T(0, 12)
S(-6, 3)
V(6, 3)
V(6, 3)
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4.5 Dilations
R(0, 0)
Your Turn
T(0, 12)
Draw R’S’T’V’
using a scale
factor of k = 1/3.
T’(0, 4)
S(-6, 3)
V(6, 3)
S’(-2, 1)
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4.5 Dilations
V’(2, 1)
R(0, 0)R’(0, 0)
Your Turn
T(0, 12)
R’S’T’V’ is a
reduction.
T’(0, 4)
S(-6, 3)
V(6, 3)
S’(-2, 1)
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4.5 Dilations
V’(2, 1)
R(0, 0)R’(0, 0)
Negative Scale Factor
In the coordinate plane, you can have
scale factors that are negative numbers.
When this occurs, the figure is dilated and
rotates 180°.
The following is still true…
 If 0 < |k| < 1 it’s a reduction.
 If |k| > 1 it’s an enlargement.
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4.5 Dilations
Example 5
Graph △FGH with vertices F(−4, −2), G(−2, 4),
and H(−2, −2) and its image after a dilation with
1
a scale factor of −
2
x, y →
1
1
−2x, −2y
F(−4, −2) →
F′(2, 1)
G(−2, 4) →
G′(1, −2)
H(−2, −2) →
H′(1, 1)
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4.5 Dilations
Your Turn
Graph △PQR with vertices P(1, 2), Q(3, 1), and R(1,
−3) and its image after a dilation with a scale factor of
−2.
x, y → −2x, −2y
P(1, 2) →
P’ (−2, −4)
Q(3, 1) →
Q’ (−6, −2)
R(1, −3) →
R’ (−2, 6)
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4.5 Dilations
Example 6
You are making your own photo stickers. Your photo is 4
inches by 4 inches. The image on the stickers is 1.1 inches by
1.1 inches. What is the scale factor of this dilation?
Image Length
S. F. =
Pre − image Length
1.1
S. F. =
4
11
S. F. =
40
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4.5 Dilations
Your Turn
An optometrist dilates the pupils
of a patient’s eyes to get a
better look at the back of the
eyes. A pupil dilates from 4.5
millimeters to 8 millimeters.
What is the scale factor of this
dilation?
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Image Length
S. F. =
Pre − image Length
8
80
S. F. =
=
4.5
45
16
S. F. =
9
4.5 Dilations
Example 7
You are using a magnifying
glass that shows the image
of an object that is six
times the object’s actual
size. Determine the length
of the image of the spider
seen through the
magnifying glass.
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Image Length
S. F. =
Pre − image Length
x
6=
1.5
x = 1.5 (6)
x = 9 cm
4.5 Dilations
Your Turn
You are using a magnifying
glass that shows the image of
an object that is six times the
object’s actual size. The
image of a spider is shown at
the left. Find the actual
length of the spider.
Image Length
S. F. =
Pre − image Length
12.6
6=
x
6
12.6
=
1
x
6x = 12.6
x = 2.1 cm
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4.5 Dilations
Summary





A dilation creates similar figures.
A dilation can be a reduction or an
enlargement.
If the scale factor is less than one, it’s a
reduction, and if the scale factor is greater than
one it’s an enlargement.
A negative scale factor is the same as a dilation
with a 180° rotation.
The scale factor is found using S. F. =
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4.5 Dilations
Image
Pre−image
Assignment
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4.5 Dilations