Dilation Formula

The Formula

From origin: (x, y) \to (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) \to (6, 4) Scale factor 0.5: (4, 6) \to (2, 3).

Notation

D_k 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

D_{k,O}: \mathbb{R}^n \to \mathbb{R}^n defined by D_{k,O}(P) = O + k(P - O) for center O and scale factor k \neq 0; d(D_{k,O}(P), D_{k,O}(Q)) = |k| \cdot 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'.

Solution

  1. 1
    Step 1: Recall the dilation rule from the origin: (x, y) \to (kx, ky) where k is the scale factor.
  2. 2
    Step 2: Apply to A(2, 4): A' = (3 \cdot 2,\, 3 \cdot 4) = (6, 12).
  3. 3
    Step 3: Apply to B(6, 0): B' = (3 \cdot 6,\, 3 \cdot 0) = (18, 0).
  4. 4
    Step 4: Apply to C(4, 8): C' = (3 \cdot 4,\, 3 \cdot 8) = (12, 24).

Answer

A'(6, 12), B'(18, 0), C'(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 = \dfrac{1}{4}. Find the image P' and compare the distance from the origin to P' vs. to P.

Common Mistakes

  • Forgetting the center of dilation — all distances are measured from this center point
  • Thinking a scale factor between 0 and 1 enlarges the figure — it actually shrinks it
  • Assuming dilation preserves distances — it preserves angles and ratios, but changes actual lengths

Why This Formula Matters

Creates similar figures; basis for scaling and proportional reasoning.

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?

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

Why is the Dilation formula important in Math?

Creates similar figures; basis for scaling and proportional reasoning.

What do students get wrong about Dilation?

Scale factor > 1 enlarges, 0 < \text{scale} < 1 shrinks, negative reflects.

What should I learn before the Dilation formula?

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

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