
A
dilation
is
a transformation (notation
) that produces an image that is the same
shape as the original, but is a different
size. A dilation stretches or shrinks
the original figure.
The description of a dilation includes the scale
factor (or ratio) and the center
of the dilation. The center of
dilation is a fixed point in the plane about which all points are
expanded or contracted. It is the only invariant point under a
dilation. 
A dilation of
scalar factor k whose center of dilation is the origin
may be written: D_{k }(x, y) = (kx, ky).
If the scale factor, k, is greater than 1, the image is
an enlargement (a stretch).
If the scale factor is between 0 and 1, the image is a reduction (a
shrink).
(It is possible, but not usual, that the scale factor is 1, thus
creating congruent figures.)
Properties preserved (invariant) under a
dilation:
1. angle measures (remain the same)
2. parallelism (parallel lines remain parallel)
3. colinearity (points stay on the same lines)
4. midpoint (midpoints remain the same in each figure)
5. orientation (lettering order remains the same)

6. distance is NOT
preserved (NOT an isometry)
(lengths of segments are NOT the same in all cases
except a scale factor or 1.) 

Dilations
create similar figures. 
Definition: A
dilation
is a transformation of the plane,
,
such that if O is a fixed
point, k is a nonzero real number, and P' is the image of
point P, then O, P and P' are collinear and
.
Notation:
Examples:
1.
P' is the image of P under a
dilation
about O of ratio 2.
OP' = 2OP and



Most dilations in
coordinate geometry use the origin, (0,0), as the center of the dilation.
Example 1:

PROBLEM: Draw the
dilation image of triangle ABC with the center of
dilation at the origin and a scale factor of 2.
OBSERVE: Notice how EVERY
coordinate of the original triangle has been multiplied by
the scale factor (x2).
HINT: Dilations involve
multiplication! 

Example 2:

PROBLEM: Draw the
dilation image of pentagon ABCDE with the center of
dilation at the origin and a scale factor of 1/3.
OBSERVE: Notice how EVERY
coordinate of the original pentagon has been multiplied by the
scale factor (1/3).
HINT: Multiplying by 1/3 is
the same as dividing by 3!


For this example, the center of the dilation is NOT the
origin. The center of dilation is a vertex of the original figure.
Example 3:

PROBLEM: Draw the
dilation image of rectangle EFGH with the center of
dilation at point E and a scale factor of 1/2.
OBSERVE:
Point E and its image are the same. It is important
to observe the distance from the center of the
dilation, E, to the other points of the figure. Notice
EF = 6 and E'F' = 3.
HINT: Be sure to measure
distances for this problem.


