Dilations
Overview
A dilation enlarges or reduces a figure with respect to a fixed center point, producing an image that is similar to — but not congruent to — the preimage. This lesson defines the scale factor as the ratio of image to preimage lengths, covers dilations centered at the origin and at any other point, and distinguishes enlargements from reductions. Watch the direction of the ratio: image divided by preimage, and the two directions are reciprocals of each other.
Vocabulary
- Dilation
- A transformation that produces an image similar to the preimage, enlarging or reducing the figure with respect to a fixed point called the center of dilation and a scale factor k. Angle measures are preserved.
- Center of Dilation
- The fixed point with respect to which a figure is enlarged or reduced. The image of each point P lies on the ray from the center through P, and the center itself maps to itself.
- Scale Factor
- The ratio of the lengths of corresponding sides of the image and the preimage: k = (image length) ÷ (preimage length). It is the same for every side of the figure.
- Enlargement
- A dilation with scale factor greater than 1, creating a proportionally larger image.
- Reduction
- A dilation with scale factor between 0 and 1, creating a proportionally smaller image.
Main Ideas
- For a dilation with center C and scale factor k, every point P maps to a point P′ on ray CP with CP′ = k · CP. The center maps to itself.
- Relative to the origin, the rule is (x, y) → (kx, ky): multiply both coordinates by the scale factor.
- For a dilation about any other point: translate so the center lands at the origin, dilate, then translate back.
- Corresponding sides of the preimage and image are proportional, all with the same ratio k. To find k from two corresponding lengths, divide the image length by the preimage length.
- A dilation preserves angle measures, so the image is similar to the preimage; it is congruent only when k = 1.
Lesson Notes
Flip through the notes one page at a time using the buttons below.
Lesson Video
Watch the lesson video.
Interactive Activity
Dilate figures from the origin and other centers, with scale factors greater than and less than 1.
Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.