Dilation Formula & Examples
To find the image of a point after dilation, apply the formula to each coordinate separately.
Example Dilations from Origin
| Original Point | Scale Factor | Dilated Point | Effect |
|---|---|---|---|
| (3, 4) | k = 2 | (6, 8) | Enlarged |
| (6, 8) | k = 0.5 | (3, 4) | Reduced |
| (2, 5) | k = −1 | (−2, −5) | Reflected |
| (4, 3) | k = 3 | (12, 9) | Enlarged 3× |
Frequently Asked Questions
Dilation is a transformation that changes the size of a figure without changing its shape. It stretches or shrinks a figure by a scale factor relative to a center point. The original figure and dilated figure are similar — all angles are preserved and sides are proportional.
The scale factor (k) determines how much a figure is enlarged or reduced. If k > 1, the figure is enlarged. If 0 < k < 1, the figure is reduced. If k = 1, the figure stays the same. If k is negative, the figure is also reflected through the center of dilation.
To dilate a point (x, y) with scale factor k about center (cx, cy): new x = cx + k×(x − cx), new y = cy + k×(y − cy). If the center is the origin (0,0), this simplifies to (kx, ky).
The center of dilation is the fixed point about which all points in the figure are dilated. Points on the figure move toward or away from this center. If the center is on the figure, that point stays fixed during the dilation.
Dilated figures are similar to the original, not congruent (unless k = 1 or k = −1). Similar figures have the same shape and proportional sides but different sizes. Congruent figures have exactly the same size and shape.
Dilation appears in photography (zooming), map scales (1 inch = 10 miles), architectural blueprints, computer graphics scaling, and medical imaging. Whenever you resize an image or scale a drawing while preserving its proportions, you are performing a dilation.