Transcript Slide 1

12-7
12-7Dilations
Dilations
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
12-7 Dilations
Warm Up
1. Translate the triangle with vertices A(2, –1),
B(4, 3), and C(–5, 4) along the vector <2, 2>.
2. Find the coordinates of the image only do
not graph.
Holt Geometry
12-7 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)
Holt Geometry
12-7 Dilations
Definition:
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. Dilation is a non-Rigid
transformation.
Holt Geometry
12-7 Dilations
Holt Geometry
12-7 Dilations
Holt Geometry
12-7 Dilations
Vocabulary
center of dilation
enlargement
reduction
Holt Geometry
12-7 Dilations
Example 1: Identifying Dilations
Tell whether each transformation appears to
be a dilation. Explain.
A.
No; the figures are not
similar.
Holt Geometry
B.
Yes; the figures are
similar and the image is
not turned or flipped.
12-7 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 Geometry
Yes, the figures are
similar and the image
is not turned or
flipped.
12-7 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 Geometry
12-7 Dilations
Holt Geometry
12-7 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 Geometry
12-7 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 Geometry
12-7 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 Geometry
W’
X’
Y’
Z’
12-7 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 Geometry
12-7 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 Geometry
12-7 Dilations
Holt Geometry
12-7 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 Geometry
12-7 Dilations
Example 3: 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 Geometry
centered at the
12-7 Dilations
Example 3: 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 Geometry
centered at the
12-7 Dilations
Example 3 Continued
Graph the preimage and image.
P
R’
Q
Holt Geometry
Q’
R
P’
12-7 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 Geometry
12-7 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 Geometry
12-7 Dilations
Check It Out! Example 4 Continued
Graph the preimage and
image.
T’
S’
U
Holt Geometry
R’
R
U’
T
S
12-7 Dilations
Homework
1. Tell whether the transformation appears to be a
dilation.
2. Copy ∆RST and the center of dilation. Draw the
image of ∆RST under a dilation with a scale of .
Holt Geometry
12-7 Dilations
Homework
3. A rectangle on a transparency has length 6cm and
width 4 cm . On the transparency 1 cm represents
12 cm on the projection. Find the perimeter of the
rectangle in the projection.
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 Geometry