Lesson 1: 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 moves each point $P$ to a point $P'$ on the ray $CP$ so that the ratio of the length of the image to the length of the original is $r$: $|CP'|/|CP| = r$.

• If $r > 1$, the figure expands.
• If $r < 1$, the figure contracts.
• If $r = 1$, the figure is unchanged.
• Lengths are multiplied by the scale factor $r$, while areas are multiplied by $r^2$.

Examples

• Dilating the point $(4, 8)$ from the origin by a factor of $r=3$ results in the new point $(4 \times 3, 8 \times 3) = (12, 24)$.
• A rectangle with sides of length 6 and 10 is dilated by a factor of $r=0.5$. The new side lengths are 3 and 5, and the new area is $15$, which is $60 \times (0.5)^2$.
• A line segment from $(1, 2)$ to $(4, 2)$ has length 3. After a dilation with factor $r=4$ from the origin, the new segment is from $(4, 8)$ to $(16, 8)$ and has length 12, which is $3 \times 4$.

Property

A dilation transforms a triangle by multiplying all side lengths by a scale factor $k$, where $k > 0$. If $k > 1$, the triangle enlarges; if $0 < k < 1$, the triangle shrinks. The dilated triangle has vertices at positions that are $k$ times the distance from the center of dilation.

Examples