Connection to previews lesson…
 Previously, we 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.
Dilations
Standard: MCC9-12.G.SRT.1 Verify experimentally the
properties of dilations given by a center and a scale factor.
EQ: What is a dilation and how does this transformation
affect a figure in the coordinate plane?
Graph:
10
A(4, 2)
B(2, 0)
5
C(6, -6)
D(0, -4)
-15
-10
-5
5
E(-6, -6)
F(-2, 0)
-5
G(-4, 2)
H(0, 4)
Connect and label “original”.
-10
10
15
10
5
H
A
G
B
F
-15
-10
-5
5
-5
E
10
D
C
-10
15
When dilating a figure you need to
have a scale factor.
For our first dilation use a scale factor
of 2.
This means you will multiply each
coordinate by 2 to get the new
location.
A(4, 2)  A’(42, 2 2)  A’(8, 4)
B(2, 0)  B’(22, 0 2)  B’(4, 0)
C(6, -6)  C’(62, -6 2)  C’(__, __)
D(0, -4) 
E(-6, -6) 
F(-2, 0) 
G(-4, 2) 
H(0, 4) 
Graph the dilation with a scale factor of 2:
A’(8, 4)
B’(4, 0)
C’(12, -12)
D’(0, -8)
E’(-12, -12)
F’(-4, 0)
G’(-8, 4)
H’(0, 8)
10
H’
5
G’
H
A’
A
G
F’
-10
-15
B
F
-5
B’
5
10
15
D
-5
E
C
D’
-10
E’
C’
Now on your graph paper calculate the coordinates for a
dilation with a scale factor of 0.5.
Here are the
original points…
A(4, 2) A”( , )
B(2, 0) B”( , )
C(6, -6) 
D(0, -4) 
E(-6, -6)
F(-2, 0) 
G(-4, 2) 
H(0, 4) 
10
H’
5
G’
G’’
F’
-10
A’
A
G
-15
H
F
H’’
A’’
F’’ B’’
B
-5
B’
5
10
15
D’’
E’’
C’’
D
-5
E
C
D’
-10
E’
C’
Vocabulary:
Dilation:
 Transformation that changes the size of a figure, but not the shape.
Scale factor:
 The ratio of any 2 corresponding lengths of the sides of 2 similar
figures.
Corresponding Sides:
 Sides that have the same relative positions in geometric figures.
Vocabulary:
Congruent:
 Having the same size, shape and measure. 2 figures are congruent
if all of their corresponding measures are equal.
Congruent figures:
 Figures that have the same size and shapes.
Corresponding Angles:
 Angles that have the same relative positions in geometric figures.
Vocabulary:
Parallel Lines:
 2 lines are parallel if they lie in the same plane and do not
intersect.
Proportion:
 An equation that states that 2 ratios are equal.
Ratio:
 Comparison of 2 quantities by division and may be written as r/s,
r:s, or r to s.
Vocabulary:
Transformation:
 The mapping or movement of all points of a figure in a plane
according to a common operation.
Similar Figures:
 Figures that have the same shape but not necessarily the same
size.
Dilation properties
•When dilating a figure in a
coordinate plane, a segment in the
original image (not passing through
the center), is parallel to it’s
corresponding segment in the
dilated image.
•When given a scale factor, the
corresponding sides of the dilated
image become larger of smaller by
the scale factor ratio given.
The center of
any dilation is
where the
lines through
all
corresponding
points
intersect.
L
C
C is the center of the
dilation mapping ΔXYZ
onto ΔLMN
Y
X
Z
M
N
Dilation types
Contraction: reduction: the image is
smaller than the preimage: scale factor is
greater than 0, but less than 1.
Expansion: enlargement: the image is larger
than preimage: Scale factor is greater than 1.
Example 1
A picture is enlarged by a scale factor of
125% and then enlarged again by the
same scale factor. If the original picture
was 4” x 6”, how large is the final copy?
By what scale factor was the original
picture enlarged?
Example 2
A triangle has coordinates A(3,-1), B(4,3)
and C(2,5). The triangle will undergo a
dilation using a scale factor of 3. Determine
the coordinates of the vertices of the resulting
triangle.
Example 3
Triangle ABC is a dilation of triangle XYZ. Use the
coordinates of the 2 triangles to determine the scale
factor of the dilation.
A(-1, 1), B(-1, 0), C(3,1)
X(-3, 3),Y(-3, 0), Z(9, 3)
Similar Figures
Two figures, F and G, are similar (written
F ~ G) if and only if
a.) corresponding angles are congruent
and
b.)corresponding sides are proportional.
Dilations always result in similar figures!!!
W
Similar Figures
If WXY ~ ABC, then:
∠W ≅ ∠A
∠X ≅ ∠B
Y
X
∠Y ≅ ∠C
A
WX XY YZ
AB BC CD
=
=
=
B
C
Example 1
If ΔABC is similar to ΔDEF in the diagram
below, then m∠D = ?
A.80°
B. 60°
C.40°
D.30°
E. 10°
B
E
80°
D
F
40°
A
C
Example 2
Determine whether the triangles are similar. Justify
your response!
13
12
9
9.75
3.75
5
Example 3
Triangle ABC is similar to triangle DEF. Determine the scale
factor of DEF to ABC (be careful – the order is important),
then calculate the lengths of the unknown sides.
A
12
B
D
15
y+3
9
C
E
y-3
x
F
Example 4
In the figure below, ΔABC is similar to ΔDEF.
What is the length of DE?
A. 12
E
B
B. 11
11
10
C. 10
C
D. 7⅓
A
D
8
12
E. 6⅔
F