Dilations

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Dilations
12-7
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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Warm Up 1. Translate the triangle with vertices
A(2, 1), B(4, 3), and C(5, 4) along the vector
lt2, 2gt.
A'(4,1), B'(6, 5),C(3, 6)
2. ?ABC ?JKL. Find the value of JK.
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Objective
Identify and draw dilations.
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Vocabulary
center of dilation enlargement reduction
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Recall that a dilation is a transformation that
changes the size of a figure but not the shape.
The image and the preimage of a figure under a
dilation are similar.
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Example 1 Identifying Dilations
Tell whether each transformation appears to be a
dilation. Explain.
A.
B.
No the figures are not similar.
Yes the figures are similar and the image is not
turned or flipped.
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Check It Out! Example 1
Tell whether each transformation appears to be a
dilation. Explain.
a.
b.
Yes, the figures are similar and the image is not
turned or flipped.
No, the figures are not similar.
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A dilation enlarges or reduces all dimensions
proportionally. A dilation with a scale factor
greater than 1 is an enlargement, or expansion. A
dilation with a scale factor greater than 0 but
less than 1 is a reduction, or contraction.
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Example 2 Drawing Dilations
Copy the figure and the center of dilation P.
Draw the image of ?WXYZ under a dilation with a
scale factor of 2.
Step 1 Draw a line through P and each vertex.
Step 2 On each line, mark twice the distance from
P to the vertex.
W
X
Step 3 Connect the vertices of the image.
Y
Z
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Check It Out! Example 2
Copy the figure and the center of dilation. Draw
the dilation of RSTU using center Q and a scale
factor of 3.
Step 1 Draw a line through Q and each vertex.
R
S
Step 2 On each line, mark twice the distance from
Q to the vertex.
Step 3 Connect the vertices of the image.
T
U
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Example 3 Drawing Dilations
On a sketch of a flower, 4 in. represent 1 in. on
the actual flower. If the flower has a 3 in.
diameter in the sketch, find the diameter of the
actual flower.
The scale factor in the dilation is 4, so a 1 in.
by 1 in. square of the actual flower is
represented by a 4 in. by 4 in. square on the
sketch.
Let the actual diameter of the flower be d in.
3 4d
d 0.75 in.
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Check It Out! Example 3
What if? An artist is creating a large painting
from a photograph into square and dilating each
square by a factor of 4. Suppose the photograph
is a square with sides of length 10 in. Find the
area of the painting.
The scale factor of the dilation is 4, so a 10
in. by 10 in. square on the photograph represents
a 40 in. by 40 in. square on the painting.
Find the area of the painting.
A l ? w 4(10) ? 4(10)
40 ? 40 1600 in2
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If the scale factor of a dilation is negative,
the preimage is rotated by 180. For k gt 0, a
dilation with a scale factor of k is equivalent
to the composition of a dilation with a scale
factor of k that is rotated 180 about the center
of dilation.
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Example 4 Drawing Dilations in the Coordinate
Plane
Draw the image of the triangle with vertices
P(4, 4), Q(2, 2), and R(4, 0) under a
dilation with a scale factor of centered
at the origin.
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Example 4 Continued
Graph the preimage and image.
P
R
Q
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Check It Out! Example 4
Draw the image of the triangle with vertices R(0,
0), S(4, 0), T(2, -2), and U(2, 2) under a
dilation centered at the origin with a scale
factor of .
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Check It Out! Example 4 Continued
Graph the preimage and image.
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Lesson Quiz Part I
1. Tell whether the transformation appears to be
a dilation.
yes
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Lesson Quiz Part II
3. A rectangle on a transparency has length 6cm
and width 4 cm and with 4 cm. On the transparency
1 cm represents 12 cm on the projection. Find the
perimeter of the rectangle in the projection.
240 cm
4. Draw the image of the triangle with vertices
E(2, 1), F(1, 2), and G(2, 2) under a dilation
with a scale factor of 2 centered at the origin.