# Where do points go in a dilation?

Dilates points in the coordinate plane by a scale factor k about any centre, P′ = C + k(P − C), and says whether the figure is enlarged or reduced, with the length and area factors.

- Page: https://www.acalculator.org/math/dilation-calculator
- JSON spec: https://www.acalculator.org/math/dilation-calculator.json
- Version: d15588678e69

## Default answer

Example with the default inputs (Scale factor k 2, Centre x 0, Centre y 0, Points [x -4, y 6; x 2, y 1; x 5, y 4]): The image points are A′(−8, 12); B′(4, 2); C′(10, 8).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| k | Scale factor k | How many times farther from the centre each image point is. A negative k also turns the figure through 180°. |
| cx | Centre x | The x coordinate of the centre of dilation; 0 when left empty. |
| cy | Centre y | The y coordinate of the centre of dilation; 0 when left empty. |
| points | Points | The points of the figure to dilate, A, B, C and so on, up to 12. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| image | Image points | Each point after the dilation, as A′(x, y); B′(x, y) and so on. |
| kind | Type of dilation | An enlargement when \|k\| > 1, a reduction when \|k\| < 1, the same size when \|k\| = 1. |
| lengthFactor | Length factor | \|k\|: every length of the image is this many times the original length. |
| areaFactor | Area factor | k²: the area of the image is this many times the original area. |
| steps | Working | For each point, x′ = cx + k(x − cx) and y′ = cy + k(y − cy). |

## Method

P′ = C + k(P − C): x′ = cx + k(x − cx), y′ = cy + k(y − cy); lengths × |k|, areas × k².

## Assumptions

- Typed decimals are read exactly, so each image coordinate is the exact decimal, rounded once to 10 significant figures.
- A negative scale factor puts the image on the other side of the centre, the same as a dilation by |k| and a half turn.

## Worked examples

1. k = 2, cx = 0, cy = 0, points = {"x":-4,"y":6} or undefined gives image = A′(−8, 12), kind = Enlargement, areaFactor = 4. Source: CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02).
2. k = 0.5, cx = 0, cy = 0, points = {"x":9,"y":-13} or undefined gives image = A′(4.5, −6.5), kind = Reduction, lengthFactor = 0.5, areaFactor = 0.25. Source: CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02).
3. k = 3, cx = 1, cy = 2, points = {"x":2,"y":3} or {"x":4,"y":2} gives image = A′(4, 5); B′(10, 2); C′(1, 11), areaFactor = 9. Source: OpenStax, Elementary Algebra 2e, §8.7 Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio), https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications (retrieved 2026-10-02).
4. k = -0.5, cx = 0, cy = 0, points = {"x":4,"y":-2} or undefined gives image = A′(−2, 1), kind = Reduction, turned 180° about the centre, areaFactor = 0.25. Source: CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the rule (x, y) → (kx, ky) about the origin; A(−4, 6) with k = 2 gives A′(−8, 12); A(9, −13) with k = 1/2 gives A′(4.5, −6.5)), https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02).

## FAQ

### What is a dilation in geometry?

A dilation resizes a figure about a fixed point, the centre of dilation, by a scale factor k. Each image point lies on the line from the centre through the original point, k times as far from the centre. The image has the same shape: it is similar to the original.

### How do I dilate a point about the origin?

Multiply both coordinates by the scale factor: (x, y) → (kx, ky). A(−4, 6) dilated by k = 2 is A′(−8, 12), and A(9, −13) dilated by k = ½ is A′(4.5, −6.5).

### How do I dilate about a point that is not the origin?

Subtract the centre, multiply by k, and add the centre back: x′ = cx + k(x − cx) and y′ = cy + k(y − cy). With centre (1, 2) and k = 3, the point (2, 3) goes to (1 + 3 × 1, 2 + 3 × 1) = (4, 5).

### Is it an enlargement or a reduction?

If |k| is more than 1 the image is bigger (an enlargement); if |k| is between 0 and 1 it is smaller (a reduction); if |k| = 1 it is the same size. A negative k also turns the image half a turn about the centre.

### What happens to lengths and areas?

Every length is multiplied by |k| and every area by k². Dilating a triangle by 3 makes each side 3 times as long and the area 9 times as large. Angles do not change.

### Can the scale factor be 0?

No. With k = 0 every point lands on the centre, so the image is a single point and not a figure. The calculator asks for a k other than 0.

## Sources

- CK-12 Foundation, Geometry, 7.16 Dilation in the Coordinate Plane (the mapping (x, y) → (kx, ky); worked examples with k = 2 and k = ½), on K12 LibreTexts. https://k12.libretexts.org/Bookshelves/Mathematics/Geometry/07:_Similarity/7.16:_Dilation_in_the_Coordinate_Plane (retrieved 2026-10-02)
- OpenStax, Elementary Algebra 2e, §8.7 Solve Proportion and Similar Figure Applications (similar figures have corresponding sides in the same ratio), CC BY 4.0. https://openstax.org/books/elementary-algebra-2e/pages/8-7-solve-proportion-and-similar-figure-applications (retrieved 2026-10-02)