Dilation Formula

Dilation is a transformation that enlarges or shrinks a figure by a scale factor from a center point.

The Formula

From origin: (x,y)→(kx,ky) where k is the scale factor

When to use: Like zooming in or out on a photo—everything gets bigger or smaller proportionally.

Quick Example

Scale factor 2 from origin: (3,2)→(6,4) Scale factor 0.5: (4,6)→(2,3).

Notation

Dk denotes dilation with scale factor k; k>1 enlarges, 0<k<1 shrinks

What This Formula Means

A transformation that enlarges or shrinks a figure by a scale factor from a center point.

Like zooming in or out on a photo—everything gets bigger or smaller proportionally.

Formal View

Dk,O:Rn→Rn defined by Dk,O(P)=O+k(P−O) for center O and scale factor k≠0; d(Dk,O(P),Dk,O(Q))=∣k∣⋅d(P,Q)

Worked Examples

Example 1

easy
Triangle ABC has vertices A(2,4), B(6,0), C(4,8). Apply a dilation from the origin with scale factor k=3. Find the image vertices A′, B′, C′.

Answer

A′(6,12), B′(18,0), C′(12,24)

First step

1
Step 1: Recall the dilation rule from the origin: (x,y)→(kx,ky) where k is the scale factor.

Full solution

  1. 2
    Step 2: Apply to A(2,4): A′=(3⋅2, 3⋅4)=(6,12).
  2. 3
    Step 3: Apply to B(6,0): B′=(3⋅6, 3⋅0)=(18,0).
  3. 4
    Step 4: Apply to C(4,8): C′=(3⋅4, 3⋅8)=(12,24).
Dilation from the origin multiplies every coordinate by the scale factor. With k=3 the triangle is enlarged to three times its original size, keeping the same shape and orientation relative to the origin.

Example 2

medium
Point P(8,12) is dilated from the origin with scale factor k=14. Find the image P′ and compare the distance from the origin to P′ vs. to P.

Example 3

easy
A triangle with side lengths 3, 4, 5 is dilated by factor 4. Find its new side lengths.

Common Mistakes

  • Adding k to coordinates instead of multiplying — from the origin, dilation multiplies: (x,y)→(kx,ky).
  • Treating the result as congruent — unless k=1, a dilation changes size, giving a similar (not congruent) figure.
  • Dilating from the wrong point — distances are scaled from the center of dilation, not always the origin.

Why This Formula Matters

Dilation is the one transformation that is NOT rigid — it is what creates similar figures rather than congruent ones. Knowing that k>1 enlarges, 0<k<1 shrinks, and the shape is preserved is the bridge from rigid motions into similarity and proportional geometry. Recognizing it by "Is the image the same shape but a scaled size, made by multiplying distances from a center by k?" — rather than by familiar numbers — is what lets a student tell it apart from translation and rotation and congruence in a mixed problem set.

Frequently Asked Questions

What is the Dilation formula?

A transformation that enlarges or shrinks a figure by a scale factor from a center point.

How do you use the Dilation formula?

Like zooming in or out on a photo—everything gets bigger or smaller proportionally.

What do the symbols mean in the Dilation formula?

Dk denotes dilation with scale factor k; k>1 enlarges, 0<k<1 shrinks

Why is the Dilation formula important in Math?

Dilation is the one transformation that is NOT rigid — it is what creates similar figures rather than congruent ones. Knowing that k>1 enlarges, 0<k<1 shrinks, and the shape is preserved is the bridge from rigid motions into similarity and proportional geometry. Recognizing it by "Is the image the same shape but a scaled size, made by multiplying distances from a center by k?" — rather than by familiar numbers — is what lets a student tell it apart from translation and rotation and congruence in a mixed problem set.

What do students get wrong about Dilation?

The procedure for dilation is the easy part; the trap is adding k to coordinates instead of multiplying. Asking "Is the image the same shape but a scaled size, made by multiplying distances from a center by k?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Dilation formula?

Before studying the Dilation formula, you should understand: transformation geo.

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