Lesson 6: Describe Dilations

Property

A dilation is a transformation that changes the size of a figure by a scale factor $k$ with respect to a fixed point called the center of dilation.

All points on the original figure are mapped to corresponding points on the image such that the distance from the center to each image point is $k$ times the distance from the center to the corresponding original point.

Property

A dilation is given by a point $C$, the center of the dilation, and a positive number $r$, the factor of the dilation. The dilation with center $C$ and factor $r$ moves each point $P$ to a point $P'$ on the ray $CP$ so that the ratio of the length of image to the length of original is $r$: $|CP'|/|CP| = r$. The coordinate rule that expresses a dilation with center the origin and factor $r$ is $(x, y) \to (rx, ry)$.

Properties of the dilation with center $C$ and factor $r$:

• If $P$ is moved to $P'$, then $|CP'|/|CP| = r$.
• If $P$ is moved to $P'$ and $Q$ is moved to $Q'$, then $|Q'P'|/|QP| = r$.
• The dilation takes parallel lines to parallel lines.
• A line and its image are parallel.
• An angle and its image have the same measure.

Examples

• A point $P(3, 5)$ is dilated from the origin with a factor of $r=2$. The new point is $P'(2 \cdot 3, 2 \cdot 5)$, which is $P'(6, 10)$.
• A line segment $AB$ has a length of 4 units. After a dilation with a factor of $r=0.5$, the new line segment $A'B'$ has a length of $0.5 \times 4 = 2$ units.
• A triangle with angles $30^\circ, 60^\circ, 90^\circ$ is dilated by a factor of 3. The new triangle's angles are still $30^\circ, 60^\circ, 90^\circ$, but its side lengths are 3 times longer.