Geometry
Dilations
Goals


Identify Dilations
Make drawings using dilations.
April 13, 2015
Rigid Transformations


Previously studied in
Chapter 7.
Rotations




Translations
These were isometries:
The pre-image and the
image were congruent.
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Dilation


Dilations are non-rigid transformations.
The pre-image and image are similar, but not
congruent.
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April 13, 2015
Dilation
rgemen
Enla
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t
Dilation
Reduction
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Dilation
R
S
C
Center of Dilation
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T
Dilation
R
2CR
CR
R
CR
S
C
Center of Dilation
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T
Dilation
R
2CR
R
CR
CS
C
Center of Dilation
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T
CR
2CS
S
CS
S
Dilation
R
2CR
R
CR
2CS
CR
S
CS
CS
C
CT
Center of Dilation
T
CT
2CT
April 13, 2015
T
S
RST ~ RST
Dilation
R
2CR
R
CR
2CS
CR
S
CS
CS
C
CT
Center of Dilation
T
CT
2CT
April 13, 2015
T
S
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 a positive number
CP'
such that k  1 and 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.
April 13, 2015
Enlargement
Dilation
R
2CR
R
CR
2CS
CR
S
CS
CS
S
C
CT
Center of Dilation
T
CT
2CT
CR ' 2CR 2


CR
CR 1
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T
Scale Factor
RST ~ R’S’T’
Dilation
R
2CR
R
CR
2CS
CR
S
CS
CS
S
C
CT
Center of Dilation
T
CT
2CT
T
CR ' CS ' CT ' R ' S ' S ' T ' R ' T '
Scale Factor:





CR CS
CT
RS
ST
RT
April 13, 2015
Example
What type of dilation is this? Reduction
G
F
F’
G’
C
K’
K
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H’
H
Example
What is the scale factor?
45
F
F’
36
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G
Notice:
15 G’
k<1
C
12
Reduction
K’
K
F ' G ' 15 1
k


FG
45 3
F ' K ' 12 1
k


FK
36 3
H’
H
Remember:

The scale factor k is
CP '
k
CP


If 0 < k < 1 it’s a reduction.
If k > 1 it’s an enlargement.
April 13, 2015
image segment
pre-image segment
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.



April 13, 2015
Graph ABC with A(1, 1), B(3, 6), C(5, 4).
B
C
Notice the
origin is here
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A
Using a scale factor of k = 2, locate points
A’, B’, and C’. P(x, y)  P’(kx, ky).
B’
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’
A
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C’
Draw ABC.
B’
C’
B
C
A’
A
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You’re done.
Notice that
rays drawn
from the
center of
dilation (the
origin)
through every
preimage
point also
passes
through the
image point.
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B’
C’
B
C
A’
A
Do this problem.
T(0, 12)
Draw RSTV
with
R(0, 0)
S(6, 3)
T(0, 12)
V(6, 3) S(-6, 3)
April 13, 2015
V(6, 3)
R(0, 0)
Do this problem.
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)
April 13, 2015
V’(2, 1)
R(0, 0)R’(0, 0)
Do this problem.
T(0, 12)
R’S’T’V’ is a
reduction.
T’(0, 4)
S(-6, 3)
V(6, 3)
S’(-2, 1)
April 13, 2015
V’(2, 1)
R(0, 0)R’(0, 0)
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.
April 13, 2015
One more time…
Image Size
Scale Factor = Pre-image Size
After
Scale Factor = Before
April 13, 2015
Enlargement or Reduction?





CP = 10 and CP’ = 20
Enlargement
What is the Scale Factor?
2
k = CP’/CP = 20/10 = 2
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Enlargement or Reduction?





CP = 150 and CP’ = 15
Reduction
What is the Scale Factor?
1/10
k = CP’/CP = 15/150 = 1/10
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Enlargement or Reduction?





CP = 20 and CP’ = 18
Reduction
What is the Scale Factor?
9/10
k = CP’/CP = 18/20 = 9/10
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Enlargement or Reduction?





CP = 15 and CP’ = 18
Enlargement
What is the Scale Factor?
6/5
k = CP’/CP = 18/15 = 6/5
April 13, 2015