Geometry:
Dilations
We have already discussed translations,
reflections and rotations. Each of these
transformations is an isometry, which means
the image is congruent
to the preimage.
In a dilation
the image is similar to the preimage.
A dilation centered at point C with a scale factor of
k, where can be defined as follows:
1. The image of point C is itself. That is, C
_____
' C
2. For any point P other than C, the
____________________________
point P' is on CP and CP'  k  CP
k  CP
NOTE: If 0  k  1 , then the dilation is a
____________
contractio n
If k  1, then the dilation is an
____________
expansion
Why is  CPQ
~  CP ' Q '
?
k  CP
AA / SAS / SSS for similarity ?
 C   C by reflexive
CP
k  CP
1
k


CQ
k  CQ
1
k
SAS
k  CQ
PQ // P ' Q '
Why ?
CAP Converse
Example: Under a dilation, triangle A(0,0), B(0,4),
C(6,0) becomes triangle A'(0,0), B'(0,10),
C'(15,0). What is the scale factor for this dilation?
By the definition , AC'  k  AC
B’(0, 10)
15  k  6
k 
15
6
C’(15, 0)
A ' B '  k  AB
Let’s consider why this theorem is true.
Since  CAB ~  CA' B' by SAS
then,
CA
CA'
k  CA

AB
A' B '

1
k
A ' B '  k  AB

AB
A' B '
Example: Line segment AB with endpoints A(2, 5)
and B(6, -1) lies in the coordinate plane. The
segment will be dilated with a scale factor of 2 5
and a center at the origin to create A ' B '. What will
be the length of A ' B '?
2  6 
AB 
A' B ' 
2
5
2
 5   1  
 2 13 
2
4
5
13
(4)  6 
2
2
52  2 13
Example: Under a dilation of scale factor 3 with the
center at the origin, what will be the coordinates of the
image of point A(3, 4)? point B(4,1)?
A ' ( 9 ,12 )
B ' (12 , 3 )
What do you notice about the coordinates of
points A and A’ as well as B and B’ in relation
to the scale factor?
Each coordinate
in the image is
3 times the correspond ing
coordinate
in the preimage.
Theorem: If the center of dilation is the origin
and the scale factor is k, the coordinates of
the point A’, the image of A(x, y), will be
__________.
(k  x, k  y )