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Dilations

A dilation resizes a figure from a fixed point, called the , by a scale factor kk. Every point moves along the ray from the center, so its distance from the center is multiplied by kk. A dilation must always be described with both pieces: the center and the scale factor.

Dilating about the origin

Need a refresher? Ratios and proportions

A ratio compares two amounts. The ratio 2 : 3 means "2 of the first for every 3 of the second."

Two ratios make a proportion when they describe the same comparison. To keep a ratio the same, multiply both parts by the same number — never add the same number to both.

For example, 2 : 3 and 6 : 9 are the same mix, because both parts were multiplied by 3. But 2 : 3 and 6 : 7 are not — adding 4 to both parts changes the mix.

Try it: Orange paint uses 2 parts red for every 3 parts yellow. If you use 6 cups of red, how many cups of yellow keep the same mix?

A scale factor behaves like a ratio: it multiplies, it never adds. When the center is the origin, the rule is simply

(x,y)↦(kx, ky)(x, y) \mapsto (kx,\, ky)

With k=2k = 2, the point (2,3)(2, 3) lands on (4,6)(4, 6) — twice as far from the origin, in the same direction. A factor k>1k > 1 enlarges the figure; a factor between 00 and 11 reduces it.

Worked example: Dilate (6, 4) about the origin by a scale factor of ½

  1. Pick the rule for a dilation centered at the origin.

    Before the next step: Which rule applies for a dilation about the origin?

Watch the side lengths in the figure above as you drag the scale factor: every length is multiplied by kk, so with k=2k = 2 each side of the image is exactly twice its preimage side.

What a dilation preserves

Make a prediction: You double a triangle with a dilation (k = 2) about the origin. What happens to its perimeter?

A dilation is not a : unless k=1k = 1, distances change, so the image is not to the preimage. But the image is always the same shape — every angle keeps its measure — which makes the image to the original.

Two more facts you can check experimentally on the figure:

  • A line that passes through the center of dilation maps onto itself.
  • A line that does not pass through the center maps to a parallel line.

Drag the scale factor and watch every side length grow or shrink.

2
A triangle ABC dilated about (0, 0) by a scale factor of 2 → triangle A′B′C′.

Practice

Score: 0/4

  1. 1.A dilation centered at the origin with scale factor 2 sends the point (2, 3) to which point?
  2. 2.A segment has length 5. After a dilation with scale factor 3, what is the length of its image?
  3. 3.A dilation with scale factor 2 preserves which of the following?
  4. 4.A dilation about the origin with scale factor k sends the point (a, b) to (ka, kb). Write an expression for the image's x-coordinate in terms of k and a.

    Any equivalent form is accepted. Use ^ for powers and * for multiply (or just write 2x).

For teachers

Standard, common misconceptions, and suggested use will live here.