Post on 11-Jan-2016
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.
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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 21, 2023
Dilation
Enlargement
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Dilation
Reduction
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Dilation
Center of Dilation
R
S
T
C
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Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
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Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
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Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
CT
CT2CT T
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Dilation
Center of Dilation
R
S
T
C
CR
CR2CR
R
CSCS
S2CS
CT
CT2CT T
RST ~ RST
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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
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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
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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
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Example
F G
HK
F’ G’
H’K’
C
What type of dilation is this? Reduction
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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
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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
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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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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’
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Draw ABC.
A
B
C
A’
B’
C’
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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.
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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)
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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)
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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)
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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.
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One more time…
Scale Factor = Image Size
Pre-image Size
Scale Factor = After
Before
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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
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