A dilation is a transformation that makes a figure larger or smaller while keeping the same shape. It is one of the main similarity transformations in geometry because it preserves angle measures and changes all lengths by the same scale factor. On the coordinate plane, dilations are easy to visualize by drawing rays from a fixed center of dilation through each point of the figure.
The image points land on those rays at distances controlled by the scale factor.
The center of dilation stays fixed, and every other point moves along a straight line through that center. If the scale factor k is greater than 1, the image is an enlargement farther from the center. If 0 < k < 1, the image is a reduction closer to the center.
When the center is the origin, the coordinate rule is especially simple: (x, y) maps to (kx, ky).
Key Facts
- A dilation maps each point P to an image point P' on the ray from the center C through P.
- Scale factor: k = CP' / CP.
- If k > 1, the dilation is an enlargement.
- If 0 < k < 1, the dilation is a reduction.
- For center at the origin: (x, y) -> (kx, ky).
- Dilations preserve angle measures and parallel lines, but multiply lengths by k and areas by k^2.
Vocabulary
- Dilation
- A dilation is a transformation that changes the size of a figure by a scale factor while keeping the same shape.
- Center of dilation
- The center of dilation is the fixed point from which distances to all points are scaled.
- Scale factor
- The scale factor is the number that tells how many times farther each image point is from the center compared with the original point.
- Image
- The image is the new figure produced after a transformation is applied to the original figure.
- Similarity
- Similarity means two figures have the same shape, equal corresponding angles, and proportional corresponding side lengths.
Common Mistakes to Avoid
- Multiplying only one coordinate by the scale factor is wrong because a dilation from the origin scales both x and y coordinates.
- Using the wrong center of dilation is wrong because the image points must lie on rays starting at the chosen center, not necessarily the origin.
- Thinking a dilation changes angle measures is wrong because dilations preserve shape and all corresponding angles remain equal.
- Confusing k with k^2 is wrong because side lengths are multiplied by k, while areas are multiplied by k^2.
Practice Questions
- 1 A triangle has vertices A(2, 1), B(4, 1), and C(2, 3). Dilate it by scale factor k = 3 centered at the origin. What are the coordinates of A', B', and C'?
- 2 Point P(8, -4) is dilated from the origin to P'(2, -1). What is the scale factor k, and is the dilation an enlargement or a reduction?
- 3 A quadrilateral is dilated from a center C with scale factor 2. Explain why the image is similar to the original, and describe what happens to its side lengths, angle measures, and area.