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)(x, y) \to (kx, ky) where kk 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)(3, 2) \to (6, 4) Scale factor 0.50.5: (4,6)→(2,3)(4, 6) \to (2, 3).

Notation

DkD_k denotes dilation with scale factor kk; k>1k > 1 enlarges, 0<k<10 < 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→RnD_{k,O}: \mathbb{R}^n \to \mathbb{R}^n defined by Dk,O(P)=O+k(P−O)D_{k,O}(P) = O + k(P - O) for center OO and scale factor k≠0k \neq 0; d(Dk,O(P),Dk,O(Q))=∣k∣⋅d(P,Q)d(D_{k,O}(P), D_{k,O}(Q)) = |k| \cdot d(P, Q)

Worked Examples

Example 1

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

Answer

A′(6,12)A'(6, 12), B′(18,0)B'(18, 0), C′(12,24)C'(12, 24)

First step

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

Full solution

  1. 2
    Step 2: Apply to A(2,4)A(2, 4): A′=(3⋅2, 3⋅4)=(6,12)A' = (3 \cdot 2,\, 3 \cdot 4) = (6, 12).
  2. 3
    Step 3: Apply to B(6,0)B(6, 0): B′=(3⋅6, 3⋅0)=(18,0)B' = (3 \cdot 6,\, 3 \cdot 0) = (18, 0).
  3. 4
    Step 4: Apply to C(4,8)C(4, 8): C′=(3⋅4, 3⋅8)=(12,24)C' = (3 \cdot 4,\, 3 \cdot 8) = (12, 24).
Dilation from the origin multiplies every coordinate by the scale factor. With k=3k=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)P(8, 12) is dilated from the origin with scale factor k=14k = \dfrac{1}{4}. Find the image P′P' and compare the distance from the origin to P′P' vs. to PP.

Example 3

easy
A triangle with side lengths 3, 4, 53,\ 4,\ 5 is dilated by factor 44. Find its new side lengths.

Common Mistakes

  • Adding kk to coordinates instead of multiplying — from the origin, dilation multiplies: (x,y)→(kx,ky)(x,y)\to(kx,ky).
  • Treating the result as congruent — unless k=1k=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>1k>1 enlarges, 0<k<10<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 kk?" — 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?

DkD_k denotes dilation with scale factor kk; k>1k > 1 enlarges, 0<k<10 < 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>1k>1 enlarges, 0<k<10<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 kk?" — 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 kk to coordinates instead of multiplying. Asking "Is the image the same shape but a scaled size, made by multiplying distances from a center by kk?" 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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