Dilation (Geometry)

Dilation (Geometry)

How to solve dilation problems: formula, examples, and their solutions.

Formula

The image of a point (x, y), whose image is under the dilation of k, is (kx, ky).

The image of a point (x, y),
whose image is under the dilation of k,
is (kx, ky).

Multiply the scale factor k on the coordinates.

Example 1

Triangle ABC has vertices A(2, 1), B(3, 2), and C(0, -2). Its image is under the dilation of 2. Find the coordinates of the vertices of the image.

Dilation of 2:
(x, y) → (2⋅x, 2⋅y)

If | k | > 1,
then the image is enlarged.

Example 2

Triangle ABC has vertices A(-4, 6), B(6, 0), and C(-3, -3). Its image is under the dilation of 1/3. Find the coordinates of the vertices of the image.

Dilation of 1/3:
(x, y) → ([1/3]⋅x, [1/3]⋅y)

If 0 < | k | < 1,
then the image is reduced.

If k = -1,
then the image shows the reflection in the origin:
(x, y) → ([-1]⋅x, [-1]⋅y) = (-x, -y).