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
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.
Holt McDougal Geometry
9-7 Dilations
Example 1: Identifying Dilations
Tell whether each transformation appears to
be a dilation. Explain.
A.
No; the figures are not
similar.
Holt McDougal Geometry
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.
Holt McDougal Geometry
Yes, the figures are
similar and the image
is not turned or
flipped.
9-7 Dilations
Helpful Hint
For a dilation with scale factor k, if k > 0, the
figure is not turned or flipped. If k < 0, the figure
is rotated by 180°.
Holt McDougal Geometry
9-7 Dilations
Holt McDougal Geometry
9-7 Dilations
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.
Holt McDougal Geometry
9-7 Dilations
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.
Holt McDougal Geometry
W’
X’
Y’
Z’
9-7 Dilations
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.
Holt McDougal Geometry
9-7 Dilations
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.
Holt McDougal Geometry
9-7 Dilations
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
Holt McDougal Geometry
9-7 Dilations
Holt McDougal Geometry
9-7 Dilations
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.
Holt McDougal Geometry
9-7 Dilations
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
9-7 Dilations
Example 4 Continued
Graph the preimage and image.
P
R’
Q
Holt McDougal Geometry
Q’
R
P’
9-7 Dilations
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
Holt McDougal Geometry
9-7 Dilations
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
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
9-7 Dilations
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.
Holt McDougal Geometry