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Geometry
Dilations
April 21, 2023
Goals
Identify Dilations Make drawings using dilations.
April 21, 2023
Rigid Transformations
Previously studied in Chapter 7.
Rotations Translations These were isometries: The pre-image and the
image were congruent.
April 21, 2023
Dilation
Dilations are non-rigid transformations. The pre-image and image are similar, but not
congruent.
April 21, 2023
April 21, 2023
Dilation
Enlargement
April 21, 2023
Dilation
Reduction
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
CT
CT2CT T
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
CT
CT2CT T
RST ~ RST
April 21, 2023
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
such that k 1 and
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.
CP'k =
CP
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
CT
CT2CT T
Enlargement
' 2 2
1
CR CR
CR CR Scale Factor
April 21, 2023
Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
CT
CT2CT T
RST ~ R’S’T’
Scale Factor:' ' ' ' ' ' ' ' 'CR CS CT R S S T R T
CR CS CT RS ST RT
April 21, 2023
Example
F G
HK
F’ G’
H’K’
C
What type of dilation is this? Reduction
April 21, 2023
Example
F G
HK
F’ G’
H’K’
C
What is the scale factor?
36 12
45
15
' ' 15
4
1
5' ' 1
3
2
36
31
F Gk
FGF K
kFK
Notice:
k < 1
Reduction
April 21, 2023
Remember:
The scale factor k is
If 0 < k < 1 it’s a reduction. If k > 1 it’s an enlargement.
'CPk
CP
image segment
pre-image segment
April 21, 2023
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 21, 2023
Graph ABC with A(1, 1), B(3, 6), C(5, 4).
A
B
C
Notice the origin is here
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Using a scale factor of k = 2, locate points A’, B’, and C’. P(x, y) P’(kx, ky).
A
B
C
A(1, 1) A’(2 1, 2 1) = A’(2, 2)
A’
B(3, 6) B’(2 3, 2 6) = B’(6, 12)
B’
C(5, 4) C’(2 5, 2 4) = C’(10, 8) C’
April 21, 2023
Draw ABC.
A
B
C
A’
B’
C’
April 21, 2023
You’re done.
A
B
C
A’
B’
C’
Notice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point.
April 21, 2023
Do this problem.
R(0, 0)
Draw RSTV with
R(0, 0)
S(6, 3)
T(0, 12)
V(6, 3) S(-6, 3)
T(0, 12)
V(6, 3)
April 21, 2023
Do this problem.
R(0, 0)
Draw R’S’T’V’ using a scale factor of k = 1/3.
S(-6, 3)
T(0, 12)
V(6, 3)
R’(0, 0)
S’(-2, 1)
T’(0, 4)
V’(2, 1)
April 21, 2023
Do this problem.
R(0, 0)
R’S’T’V’ is a reduction.
S(-6, 3)
T(0, 12)
V(6, 3)
R’(0, 0)
S’(-2, 1)
T’(0, 4)
V’(2, 1)
April 21, 2023
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 21, 2023
One more time…
Scale Factor = Image Size
Pre-image Size
Scale Factor = After
Before
April 21, 2023
Enlargement or Reduction?
CP = 10 and CP’ = 20 Enlargement What is the Scale Factor? 2 k = CP’/CP = 20/10 = 2
April 21, 2023
Enlargement or Reduction?
CP = 150 and CP’ = 15 Reduction What is the Scale Factor? 1/10 k = CP’/CP = 15/150 = 1/10
April 21, 2023
Enlargement or Reduction?
CP = 20 and CP’ = 18 Reduction What is the Scale Factor? 9/10 k = CP’/CP = 18/20 = 9/10
April 21, 2023
Enlargement or Reduction?
CP = 15 and CP’ = 18 Enlargement What is the Scale Factor? 6/5 k = CP’/CP = 18/15 = 6/5