Transcript 3.7 Dilations

Example
• A quadrilateral ABCD has vertices
A = (-7,4) B = (0, 3) C = (5, 1),
and D = (-2, 2).
• It is translated by the vector  5,1 .
Graph ABCD and its translation
image A’B’C’D’ on the coordinate
plane.
Dilations
Definitions
• Scale Factor: The scale factor is the
number we multiply the sides or
coordinates of a figure by to get the new
coordinates or sides.
• Enlargement: When you multiply by a
number whose absolute value is greater
then 1. (Gets Bigger)
• Reduction: When you multiply by a
number whose absolute value is less
than 1. (Gets Smaller)
Identifying Dilations
Previously, you studied rigid transformations, in which the image and
preimage of a figure are congruent. In this lesson, you will study a type of
nonrigid transformation called a dilation, in which the image and preimage of
a figure are similar.
A dilation with center C and scale factor k is a transformation that maps every
point P in the plane to a point P so that the following properties are true.
If P is not the center point C, then the image point P lies on CP .
The scale factor k is a positive number such that k = CP  , and k  1.
CP
If P is the center point C, then P = P .
Identifying Dilations
The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1.
P
P´
6
P´
2
3
C
•
5
P
Q
Q´
C
R
•
R´
Q
R´
Reduction: k =
Because
PQR ~
Q´
CP
3
1
= =
CP
6
2
P´Q´R´,
R
Enlargement: k =
CP
5
=
CP
2
P´Q´
is equal to the scale factor of the dilation.
PQ
Identifying Dilations
Identify the dilation and find its scale factor.
3
2
C
P´
P
•
SOLUTION
Because CP = 2 , the scale factor
CP 3
is k = 2 . This is a reduction.
3
Identifying Dilations
Identify the dilation and find its scale factor.
3
2
C
P´
P
P´
2
P
1
•
•C
SOLUTION
SOLUTION
Because CP = 2 , the scale factor
CP 3
is k = 2 . This is a reduction.
3
Because CP = 2 , the scale factor
CP 1
is k = 2. This is an enlargement.
Dilation in a Coordinate Plane
In a coordinate plane, dilations whose centers are the origin have the
property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).
1
Use the origin as the center and use a scale factor of . How does the
2
perimeter of the preimage compare to the perimeter of the image?
SOLUTION
Because the center of the dilation is the origin,
you can find the image of each vertex by
multiplying its coordinates by the scale factor.
y
D
A(2, 2)  A´(1, 1)
D´ A
B(6, 2)  B´(3, 1)
1
C(6, 4)  C ´(3, 2)
O
D(2, 4)  D´(1, 2)
C
C´
B
A´
•
B´
1
x
Dilation in a Coordinate Plane
In a coordinate plane, dilations whose centers are the origin have the
property that the image of P(x, y) is P´(kx, ky).
Draw a dilation of rectangle ABCD with A(2, 2), B(6, 2), C(6, 4), and D(2, 4).
1
Use the origin as the center and use a scale factor of . How does the
2
perimeter of the preimage compare to the perimeter of the image?
SOLUTION
From the graph, you can see that the preimage
has a perimeter of 12 and the image has a
perimeter of 6.
A preimage and its image after a dilation are
similar figures.
Therefore, the ratio of the perimeters of a
preimage and its image is equal to the scale
factor of the dilation.
y
D
D´ A
C´
B
A´
1
O
C
•
B´
1
x
Using Dilations in Real Life
Shadow puppets have been used in many countries for hundreds of
years. A flat figure is held between a light and a screen. The audience on
the other side of the screen sees the puppet’s shadow. The shadow is a
dilation, or enlargement, of the shadow puppet. When looking at a cross
sectional view, LCP ~ LSH.
The shadow puppet shown is 12 inches tall (CP in the diagram). Find the height
of the shadow, SH, for each distance from the screen. In each case, by what
percent is the shadow larger than the puppet?
LC = LP = 59 in.; LS = LH = 74 in.
SOLUTION
59
12
=
74 SH
59 (SH) = 888
SH  15 inches
LC CP
=
LS SH
Using Dilations in Real Life
Shadow puppets have been used in many countries for hundreds of
years. A flat figure is held between a light and a screen. The audience on
the other side of the screen sees the puppet’s shadow. The shadow is a
dilation, or enlargement, of the shadow puppet. When looking at a cross
sectional view, LCP ~ LSH.
LC = LP = 59 in.; LS = LH = 74 in.; SH  15 inches
SOLUTION
To find the percent of size increase, use the scale
factor of the dilation.
scale factor =
SH
CP
15
= 1.25
12
So, the shadow is 25 % larger than the puppet.
Using Dilations in Real Life
Shadow puppets have been used in many countries for hundreds of
years. A flat figure is held between a light and a screen. The audience on
the other side of the screen sees the puppet’s shadow. The shadow is a
dilation, or enlargement, of the shadow puppet. When looking at a cross
sectional view, LCP ~ LSH.
LC = LP = 66 in.; LS = LH = 74 in.
SOLUTION
66
12
=
74 SH
66 (SH) = 888
SH  13.45 inches
LC CP
=
LS SH
Using Dilations in Real Life
Shadow puppets have been used in many countries for hundreds of
years. A flat figure is held between a light and a screen. The audience on
the other side of the screen sees the puppet’s shadow. The shadow is a
dilation, or enlargement, of the shadow puppet. When looking at a cross
sectional view, LCP ~ LSH.
LC = LP = 66 in.; LS = LH = 74 in.; SH  13.45 inches
SOLUTION
To find the percent of size increase, use the scale
factor of the dilation.
scale factor =
SH
CP
13.45
= 1.12
12
So, the shadow is 12 % larger than the puppet.
Example of a dilation
• The dilation is a reduction if 0<k<1 and it is an
enlargement if k > 1
5
2
1
2
Example 1
• Identify the dilation and find its scale factor
C
Example 1
• Identify the dilation and find its scale factor
C
Reduction, Because
CP ' 2

CP 3
The scale factor is k = ⅔
Enlargement, Because
CP ' 2

CP 1
The scale factor is k = 2
Example 2
Dilation in a Coordinate Plane
• Draw a dilation of a rectangle ABCD with vertices A(1,1),
B(3,1), C(3,2) and D(1,2). Use the origin as the center
and use a scale factor of 2. How does the perimeter of
the preimage compare to the perimeter of the image?
Example 2
Dilation in a Coordinate Plane
•
Draw a dilation of a rectangle ABCD with vertices A(1,1),
B(3,1), C(3,2) and D(1,2). Use the origin as the center
and use a scale factor of 2. How does the perimeter of
the preimage compare to the perimeter of the image?
• Preimage has a perimeter of 6
• Image has a perimeter of 12
• The perimeter was enlarged by a
factor of 2