Lesson 7.1.3

Dilations & Scale Factor

A dilation is a transformation that changes the size of a figure but preserves its shape. The scale factor controls how much bigger or smaller the image becomes.

Introduction

Unlike rigid motions (translations, reflections, rotations), a dilation can change the size of a figure. Every dilation has a center of dilation and a scale factor . Points move along rays from the center — farther if , closer if .

Past Knowledge

Transformations (Unit 4). Ratios & proportions (7.1.1–7.1.2).

Today's Goal

Perform dilations, calculate scale factors, and classify as enlargement or reduction.

Future Success

Coordinate dilations (7.1.4), similarity (7.2), scale models.

Key Concepts

Dilation

A dilation with center and scale factor maps every point to a point on ray such that:

Scale Factor Classification

→ Enlargement (image is bigger)
→ Reduction (image is smaller)
→ Identity (same size, same position)
→ Image is on the opposite side of center (also includes a 180° rotation)

Key Properties

Angles are preserved (congruent)
Side lengths are multiplied by
Perimeter is multiplied by
Area is multiplied by
The image is similar to the pre-image

Worked Examples

Basic

Enlargement (k = 2)

A triangle is dilated from center with . If side , find . Is this an enlargement or reduction?

Since , this is an enlargement.

— an enlargement

Intermediate

Reduction (k = ½)

A rectangle with perimeter 24 is dilated with . Find the perimeter and area of the image if the original area is 32.

New perimeter =

New area =

Perimeter = 12, Area = 8

Advanced

Finding the Scale Factor

has sides 6, 8, and 10. Its image has a longest side of 25. Find and all image side lengths.

Longest original side = 10, longest image side = 25

Image sides: , ,

, sides: 15, 20, 25

Common Pitfalls

Thinking Dilations Change Angles

Dilations preserve all angle measures. A 90° angle stays 90° no matter the scale factor. Only lengths change.

Using k for Area Instead of k²

If , the new area is the original (not ). Area scales by , not .

Real-Life Applications

Photocopier Zoom

Setting a copier to 150% is a dilation with . Setting it to 50% is . The center of dilation is roughly the center of the glass.

Scale Models

A 1:100 model of a building uses . Every dimension is 1/100 of the real building, but all angles and proportions are preserved.