Dilation

Geometry
definition

Also known as: scaling, enlargement, shrinking

Grade 6-8

View on concept map

A transformation that enlarges or shrinks a figure by a scale factor from a center point. Creates similar figures; basis for scaling and proportional reasoning.

Definition

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

💡 Intuition

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

🎯 Core Idea

Dilation changes size but preserves shape and angle measures.

Example

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

Formula

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

Notation

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

🌟 Why It Matters

Creates similar figures; basis for scaling and proportional reasoning.

💭 Hint When Stuck

Draw a line from the center of dilation through each vertex. Multiply each distance by the scale factor to find the new point.

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)

🚧 Common Stuck Point

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

⚠️ 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

Frequently Asked Questions

What is Dilation in Math?

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

Why is Dilation important?

Creates similar figures; basis for scaling and proportional reasoning.

What do students usually get wrong about Dilation?

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

What should I learn before Dilation?

Before studying Dilation, you should understand: transformation geo.

Prerequisites

transformation geo

How Dilation Connects to Other Ideas

To understand dilation, you should first be comfortable with transformation geo. Once you have a solid grasp of dilation, you can move on to similarity, scaling in space and proportional geometry.

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Interactive Playground

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