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.

What is the Dilation formula?

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

When do you use Dilation?

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

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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