Stretching and shrinking
A dilation changes the size of a figure without changing its shape. Every point moves along a line from the centre of dilation, and its distance from the centre is multiplied by the scale factor k.
| k | Result |
|---|---|
| k > 1 | enlargement |
| k = 1 | same size |
| 0 < k < 1 | reduction |
| k < 0 | the image flips to the other side of the centre (and is scaled by the absolute value of k) |
The image is always similar to the original: angles stay the same and sides are proportional.
Dilation about the origin
Multiply both coordinates by k:
(x, y) → (kx, ky)
Example: Triangle A(1, 2), B(3, 1), C(2, 4), scale factor 3.
- A′(3, 6), B′(9, 3), C′(6, 12)
Example: Square with corners (4, 4), (8, 4), (8, 8), (4, 8), scale factor ½.
- (2, 2), (4, 2), (4, 4), (2, 4)
Dilation about another point
If the centre is (a, b):
- Find the vector from the centre to the point.
- Multiply it by k.
- Add it back to the centre.
Example: Dilate P(5, 3) by k = 2 about C(1, 1).
- From C to P: (4, 2)
- Times 2: (8, 4)
- P′ = (1 + 8, 1 + 4) = (9, 5)
What changes and what doesn’t
| Property | After a dilation |
|---|---|
| Angles | unchanged |
| Side lengths | multiplied by k |
| Perimeter | multiplied by k |
| Area | multiplied by k² |
| Parallel lines | stay parallel |
| Orientation | same (for k > 0) |
Finding the scale factor
k = image length ÷ original length.
If a side of 6 becomes 15, k = 15 ÷ 6 = 2.5.
Real-life dilations
- Zooming a photo on a phone
- Projecting a slide on a screen
- Enlarging blueprints on a photocopier
- The pupil of your eye dilating in the dark
Worked examples
Example 1: Dilate (−2, 6) by k = ½ about the origin → (−1, 3).
Example 2: A shape’s area is 10 cm². After a dilation with k = 3, its area is? 10 × 9 = 90 cm².
Example 3: A segment of length 12 becomes 4. Scale factor? ⅓ (reduction).
Common mistakes
- Adding k instead of multiplying.
- Forgetting to scale from the centre when it isn’t the origin.
- Scaling area by k instead of k².
Try it yourself
- Dilate (3, −4) by k = 2 about the origin.
- Dilate (10, 5) by k = 0.4 about the origin.
- A triangle’s perimeter is 18 cm. After k = 1.5, perimeter?
- Dilate (4, 7) by k = 3 about (1, 1).
Answers: 1) (6, −8) 2) (4, 2) 3) 27 cm 4) (10, 19)
Quick recap
- A dilation scales distances from a centre of dilation by scale factor k.
- About the origin: (x, y) → (kx, ky).
- The image is similar; lengths × k, area × k².