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Dilations Determine whether a dilation is an enlargement, a
reduction,or a congruence transformation. Determine the scale
factor for a given dilation.Vincent Van Goghs Starry Night reduced
by a factor of 2
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CLASSIFY DILATIONSA dilation is a transformation that may change
the size of a figure.A dilation requires a center point and a scale
factor.
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CLASSIFY DILATIONSA dilation is a transformation that may change
the size of a figure.A dilation requires a center point and a scale
factor.Example:ABCDABDr = 2CA = 2(CA)CB = 2(CB)CD = 2(CD)center
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CLASSIFY DILATIONSA dilation is a transformation that may change
the size of a figure.A dilation requires a center point and a scale
factor.Another example:r = 1/3MNPOXcenterMPNOXM = (XM)XN = (XN)XO =
(XO)XP = (XP)
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Key ConceptDilationIf |r| > 1, the dilation is an
enlargement.If 0 < |r| < 1, the dilation is a reduction.If
|r| = 1, the dilation is a congruence transformation.
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Key ConceptDilationWe have seen that dilations produce similar
figures, therefore dilation is a similarity
transformation.ABCDABDandMNPOXMPNO
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TheoremIf a dilation with center C and a scale factor of r
transforms A to E and B to D, then ED = |r|(AB)ABDEC
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Example 1 Determine Measures Under DilationsFind the measure of
the dilation image AB or the preimage AB using the given scale
factor.a.AB = 12, r = -2AB = |r|(AB)AB = 2(12)AB = 24b.AB = 36, r =
AB = |r|(AB)36 = (AB)AB = 144When the scale factor is negative, the
image falls on the opposite side of the center than the
preimage.
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Key ConceptDilationsIf r > 0, P lies on CP, and CP = r CPIf r
< 0, P lies on CP the ray opposite CP, and CP = r CPThe center
of dilation is always its own image.
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Example 2 Draw a DilationDraw the dilation image of JKL with
center C and r = JKL
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Example 2 Draw a DilationDraw the dilation image of JKL with
center C and r = - JKLCJKLJLKSince 0 < |r| < 1, the dilation
is a reduction
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TheoremIf P(x, y) is the preimage of a dilation centered at the
origin with scale factor r, then the image is P(rx, ry).
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Example 3 Dilations in the Coordinate PlaneCOORDINATE GEOMETRY
Triangle ABC has verticesA(7, 10), B(4, -6), and C(-2, 3). Find the
image of ABC after a dilation centered at the origin with the scale
factor of 2. Sketch the preimage and the
image.222018161412108642-2-4-6-8-10-12-10-551015
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Example 3 Dilations in the Coordinate PlaneCOORDINATE GEOMETRY
Triangle ABC has verticesA(7, 10), B(4, -6), and C(-2, 3). Find the
image of ABC after a dilation centered at the origin with the scale
factor of 2. Sketch the preimage and the
image.222018161412108642-2-4-6-8-10-12-10-551015
Preimage (x, y)Image(2x, 2y)A(7, 10)A(14, 20)B(-4, 6)B(8,
-12)C(-2, 3)C(-4, 6)
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Example 4 Identify the Scale FactorDetermine the scale factor
for each dilation with center C. Then determine whether the
dilation is an enlargement, reduction, or congruence
transformation.a.DEBACABEDScale factor =image lengthpreimage
length
= 6 units 3 units
= 2
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Example 4 Identify the Scale FactorDetermine the scale factor
for each dilation with center C. Then determine whether the
dilation is an enlargement, reduction, or congruence
transformation.a.JFHCGScale factor =image lengthpreimage length
= 4 units 4 units
= 1The dilation is a congruence transformation
