Dilations

Warm-up

Three quick questions from earlier lessons.

Discover

A photo is enlarged so every length doubles. What happens to the angles in the photo?

They stay the same
They double

Change the scale factor k. What happens to the angle at A′?

It stays 90°
It is multiplied by k too

Key idea

A dilation stretches from a center.

A dilation with center

O

and scale factor

k > 0

sends each point

P

to

P'

on ray

OP

with

OP' = k \cdot OP

. About the origin:

(x, y) \to (kx, ky)

. Lengths are multiplied by

k

; angles are unchanged.

Worked example

Dilate $A(2, -1)$ about the origin with $k = 3$ .

Step 1Triangle ABC and the center O.
Step 2Each point slides along its ray from O.
Step 3k = 2: every distance from O doubles; the angles do not change.

A' = (6, -3)

Multiply both coordinates.

OA' = 3 \cdot OA

Check the distance.

Your turn, guided

Your turn: a segment of 4 becomes 10. The scale factor.

2.5

10 ÷ 4.

Practice ★

Dilate $A(3, 1)$ about the origin with scale factor 2. Tap its image.
Dilate $A(-3, 1)$ about the origin with scale factor 3. Tap its image.
Dilate $A(-3, -1)$ about the origin with scale factor 2. Tap its image.

Watch out!

Practice ★★

A dilation takes a segment of length 3 to a segment of length 11. Find the scale factor.
A dilation takes a segment of length 3 to a segment of length 26. Find the scale factor.

A triangle is dilated with scale factor 5. What happens to its angles?
1. They stay the same
2. They are multiplied by 5
3. They grow by 5 degrees

Exit ticket

Three questions to check you have got it.

Dilate $A(3, 1)$ about the origin with scale factor 2. Tap its image.
Dilate $A(-3, 1)$ about the origin with scale factor 3. Tap its image.
Dilate $A(-3, -1)$ about the origin with scale factor 2. Tap its image.

Summary card

Dilations

$(x, y) \to (kx, ky)$ about the origin; lengths × $k$ , angles kept.

Watch out: multiply, do not add.

About this lesson