9-6 Dilations
You identified dilations and verified them as
similarity transformations.
• Draw dilations.
• Draw dilations in the coordinate plane.
Definitions
A dilation or scaling is a similarity transformation that enlarges
or reduces a figure proportionally with respect to a center point
and a scale factor.
Scale factor is the amount of enlargement or reduction needed
to get one figure from the other when two figures are similar.
This is the number you multiply the side lengths of one figure by
to create a second similar figure.
p. 674
Scale Factor
If the scale factor is
greater than 1, the
similar figure is an
enlargement.
If the scale factor is
less than 1, it is a
reduction.
Scale Factors
Draw a Dilation
Copy trapezoid PQRS and point C.
Then use a ruler to draw the image of
trapezoid PQRS under a dilation with
center C and scale factor 3.
Since k > 1, the dilation is an enlargement of
trapezoid PQRS.
Copy trapezoid PQRS and point C. Then
use a ruler to draw the image of trapezoid
PQRS under a dilation with center C and
scale factor 3.
Locate P', Q', R', and S' so that
Copy trapezoid PQRS and point C. Then
use a ruler to draw the image of
trapezoid PQRS under a dilation with
center C and scale factor 3.
Which diagram shows the dilation image of ΔLMN
with center C and
?
A.
B.
C.
D.
PUPPETS To create the illusion of a “life-sized”
image, puppeteers sometimes use a light source to
show an enlarged image of a puppet projected on a
screen or wall. Suppose that the distance between
a light source L and the puppet is 24 inches (LP). To
what distance PP' should
you place the puppet from
the screen to create a
49.5-inch tall shadow (I'M')
from a 9-inch puppet?
Understand
This problem involves a dilation. The
center of the dilation is L, LP = 24 in.,
IM = 9 in., I'M' = 49.5 in. You are
asked to find PP'.
Plan
Find the scale factor of the dilation
from the preimage IM to the image
I'M'. Use the scale factor to find LP
and then use LP and LP' to find PP'.
Solve
The scale factor k of the enlargement is the
ratio of the length on the image to a
corresponding length on the preimage.
Scale factor of image
image = I'M', preimage = IM
Divide.
Use this scale factor of 5.5 to find LP'.
LP' = k(LP)
Definition of dilation
= 5.5(24)
k = 5.5 and LP = 24
= 132
Multiply.
Use LP' and LP to find PP'.
LP + PP' = LP'
Segment Addition
Use LP' and LP to find PP'.
LP + PP'
=
LP'
Segment Addition
24 + PP'
=
132
LP = 24 and LP' = 132
PP'
108
Subtract 24 from each side.
=
Answer: So, the puppet should be placed so that the
distance from it to the screen (PP') is 108
inches.
Check Since the dilation is an enlargement, the
scale factor should be greater than 1. Since
5.5 > 1, the scale factor is reasonable.
Use this rule to find the image of a figure after a
dilation centered at the origin (0,0).
p. 676
Trapezoid EFGH has vertices E(–8, 4), F(–4, 8),
G(8, 4) and H(–4, –8). Graph the image of EFGH after a
dilation centered at the origin with a scale factor of ¼
Multiply the x- and y-coordinates of each vertex by the scale
factor, ¼
Graph the preimage and image.
Answer: E'(–2, 1), F'(–1, 2), G'(2, 1), H'(–1, –2)
Triangle ABC has vertices A(–1, 1), B(2, –2), and
C(–1, –2). Find the image of ΔABC after a dilation
centered at the origin with a scale factor of 2. Sketch
the preimage and the image.
A.
B.
C.
D. none of the above