 |
 |
|
Math
A |
Dilations |
|
For an
intuitive lesson on dilations, see
An
Intuitive
Notion of Dilation.
Remember:
|
A
dilation
is
a transformation that produces an image that is the same
shape as the original, but is a different
size. The description of a dilation includes the scale
factor and the center of the dilation.
A dilation of
scalar factor k whose center of dilation is the origin
may be written: Dk(x,y) = (kx,ky).
If the scale factor is greater than 1, the image is
an enlargement.
If the scale factor is between 0 and 1, the image is a reduction.
|
 |
You are
probably familiar with the word "dilate"
as it relates to the eye. The pupil of the eye dilates
(gets larger or smaller) depending upon the amount of
light striking the eye. |
This same
concept of
"getting larger or smaller"
also applies to the
word dilate used in mathematics. Let's examine some dilations
related to coordinate geometry.
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.
|
|
Dilations
always involve a change in size.
|
