Transcript Slide 1

9-7
9-7 Dilations
Dilations
Warm Up
Lesson Presentation
Lesson Quiz
Holt
HoltMcDougal
GeometryGeometry
9-7 Dilations
Warm Up
1. Translate the triangle with vertices A(2, –1),
B(4, 3), and C(–5, 4) along the vector <2, 2>.
A'(4,1), B'(6, 5),C(–3, 6)
2. ∆ABC ~ ∆JKL. Find the value of JK.
Holt McDougal Geometry
9-7 Dilations
Objective
Identify and draw dilations.
Holt McDougal Geometry
9-7 Dilations
Vocabulary
center of dilation
enlargement
reduction
Holt McDougal Geometry
9-7 Dilations
A dilation is a transformation that changes the
size of a figure but not its shape. The preimage
and the image are always similar.
A
Holt McDougal Geometry
A’
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Example 1: Identifying Dilations
Tell whether each transformation appears to
be a dilation. Explain.
A.
No; the figures are not
similar.
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B.
Yes; the figures are
similar and the image is
not turned or flipped.
9-7 Dilations
Check It Out! Example 1
Tell whether each transformation appears to
be a dilation. Explain.
a.
b.
No, the figures are
not similar.
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Yes, the figures are
similar and the image
is not turned or
flipped.
9-7 Dilations
Center of
dilation
K=
CP
CQ
PP’
QQ’
PQ
P’Q’
A dilation, or similarity transformation, is a
transformation in which every point P and its image P’
have the same ratio.
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9-7 Dilations
A scale factor describes how much the figure is
enlarged or reduced.
For a dilation with scale factor k, you can find
the image of a point by multiplying each
coordinate by k: (a, b)  (ka, kb).
k > 1 is an enlargement, or expansion.
0< k < 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.
Step 3 Connect the
vertices of the image.
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W
’
X’
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’
U’
T’
Step 2 On each line,
mark twice the
distance from Q to
the vertex.
Step 3 Connect the
vertices of the image.
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9-7 Dilations
Example 1: Drawing and Describing Dilations
A. Apply the dilation D to the polygon with the
given vertices. Describe the dilation.
D: (x, y) → (3x, 3y)
A(1, 1), B(3, 1), C(3, 2)
A’ (3, 3), B’ (9, 3), C’ (9,6)
Holt McDougal Geometry
scale factor 3
9-7 Dilations
Example 1: Continued
B. Apply the dilation D to the polygon with the
given vertices. Describe the dilation.
D: (x, y) → 3 x, 3 y
4
4
P(–8, 4), Q(–4, 8), R(4, 4)
P’(-6, 3), Q’ (-3, 6), R’ (3, 3)
Holt McDougal Geometry
scale factor 3/4
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Check It Out! Example 1
Name the coordinates of the image points.
Describe the dilation.
(x, y)→ ( ¼ x, ¼ y)
D(-8, 0), E(-8, -4), and F(-4, -8).
D'(-2, 0), E'(-2, -1), F'(-1, -2); scale factor 1/4
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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 > 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
origin.
The dilation of (x, y) is
Holt McDougal Geometry
centered at the
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Example 4 Continued
Graph the preimage and image.
P
R’
Q
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Q’
R
P’
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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
.
The dilation of (x, y) is
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Check It Out! Example 4 Continued
Graph the preimage and
image.
T’
S’
U
Holt McDougal Geometry
R
’R
U’
T
S
9-7 Dilations
Determine whether
the polygons are
similar.
A(–6, -6), B(-6, 3),
C(3, 3), D(3, -6)
H(-2, -2), J(-2, 1),
K(1, 1), L(1, -2)
ABCD maps to HJKL
1 x, 1 y
(x, y) →
3
3
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Determine whether
the polygons are
similar.
P(2, 0), Q(2, 4),
R(4, 4),S(4, 0)
W(5, 0), X(5, 10),
Y(8, 10), Z(8, 0).
No;
(x, y) → (2.5x, 2.5y)
maps P to W,
but not S to Z.
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Determine whether
the polygons are
similar.
A(1, 2), B(2, 2), C(1, 4)
D(4, -6), E(6, -6), F(4, -2)
Yes;
translation:
(x, y) → (x + 1, y - 5).
dilation:
(x, y) → (2x, 2y).
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Determine whether
the polygons are
similar.
F(3, 3), G(3, 6),
H(9, 3), J(9, –3)
S(–1, 1), T(–1, 2),
U(–3, 1), V(–3, –1)
Yes;
reflection:
(x, y) → (-x, y).
dilation:
(x, y) → (1/3 x, 1/3 y)
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Determine whether
the polygons are
similar.
A(2, -1), B(3, -1), C(3, -4)
P(3, 6), Q(3, 9), R(12, 9).
Yes;
rotation:
(x, y) → (-y, x)
dilation:
(x, y) → (3x, 3y)
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Determine whether
the polygons are
similar. If so, describe
the transformation in
2 different ways, from
the larger to the
smaller, and the
smaller to the larger.
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Determine whether
the polygons are
similar. If so,
describe the
transformation in 2
different ways,
from the larger to
the smaller, and
the smaller to the
larger.
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Example 3: Proving Triangles Are Similar
Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2),
and J(6, 2).
Prove: ∆EHJ ~ ∆EFG.
Step 1 Plot the points
and draw the triangles.
Holt McDougal Geometry
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Example 3 Continued
Step 2 Use the Distance Formula to find the side lengths.
Holt McDougal Geometry
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Example 3 Continued
Step 3 Find the similarity ratio.
=2
Since
=2
and E  E, by the Reflexive Property,
∆EHJ ~ ∆EFG by SAS ~ .
Holt McDougal Geometry
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Check It Out! Example 3
Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and
V(4, 3).
Prove: ∆RST ~ ∆RUV
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Check It Out! Example 3 Continued
Step 1 Plot the points and draw the triangles.
Y
5
4
U
V
3
2
X
S
1
-7 -6 -5 -4 -3 -2 -1
R
-1
Holt McDougal Geometry
T
1
2
3
4
5
6
7
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Check It Out! Example 3 Continued
Step 2 Use the Distance Formula to find the side lengths.
Holt McDougal Geometry
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Check It Out! Example 3 Continued
Step 3 Find the similarity ratio.
Since
and R  R, by the Reflexive
Property, ∆RST ~ ∆RUV by SAS ~ .
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Example 4: Using the SSS Similarity Theorem
Graph the image of ∆ABC
after a dilation with scale
factor
Verify that ∆A'B'C' ~ ∆ABC.
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Example 4 Continued
Step 1 Multiply each coordinate by
to find the
coordinates of the vertices of ∆A’B’C’.
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Example 4 Continued
Step 2 Graph ∆A’B’C’.
B’ (2, 4)
A’ (0, 2)
C’ (4, 0)
Holt McDougal Geometry
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Example 4 Continued
Step 3 Use the Distance Formula to find the side lengths.
Holt McDougal Geometry
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Example 4 Continued
Step 4 Find the similarity ratio.
Since
Holt McDougal Geometry
, ∆ABC ~ ∆A’B’C’ by SSS ~.
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Check It Out! Example 4
Graph the image of ∆MNP
after a dilation with scale
factor 3.
Verify that ∆M'N'P' ~ ∆MNP.
Holt McDougal Geometry
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Check It Out! Example 4 Continued
Step 1 Multiply each coordinate by 3 to find the
coordinates of the vertices of ∆M’N’P’.
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Check It Out! Example 4 Continued
Step 2 Graph ∆M’N’P’.
Y
7
6
5
4
3
2
1
Holt McDougal Geometry
1
2
3
4
5
6
7
X
-7 -6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-7
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Check It Out! Example 4 Continued
Step 3 Use the Distance Formula to find the side lengths.
Holt McDougal Geometry
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Check It Out! Example 4 Continued
Step 4 Find the similarity ratio.
Since
Holt McDougal Geometry
, ∆MNP ~ ∆M’N’P’ by SSS ~.
9-7 Dilations
Lesson Quiz: Part I
1. Tell whether the transformation appears to be a
dilation.
yes
2. Copy ∆RST and the center of dilation. Draw the
image of ∆RST under a dilation with a scale of .
Holt McDougal Geometry
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Lesson Quiz : Part-I
3. Apply the dilation D: (x, y)
to the
polygon with vertices A(2, 4), B(2, 6), and
C(6, 4). Name the coordinates of the image
points. Describe the dilation.
A’(3, 6), B’(3, 9), C’(9, 6); k= 3/2
4. Given X(0, 2), Y(–2, 2), and Z(–2, 0), find the
dilation with scale factor –4.
X'(0, –8); Y'(8, –8); Z'(8, 0)
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Lesson Quiz : Part-II
Determine whether the polygons with the given
vertices are similar.
5. A(-4, 4), B(6, 4), C(6, -4), D(-4, -4) and
P(-2, 2), Q(4, 2), R(4, -2), S(-2, -2)
No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q.
6. A(2, 2), B(2, 4), C(6, 4) and
D(3, -3), E(3, -6), F(9, -6)
Yes; △ ABC maps to △ A’B’C’ by a reflection:
(x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a
dilation:(x, y) → (1.5x, 1.5y).
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Lesson Quiz: Part II
3. Given: A(–1, 0), B(–4, 5), C(2, 2), D(2, –1),
E(–4, 9), and F(8, 3)
Prove: ∆ABC ~ ∆DEF
Therefore,
by SSS ~.
Holt McDougal Geometry
and ∆ABC ~ ∆DEF