## The Lesson

A shape can be enlarged with a negative scale factor. To describe an enlargement, we need to describe the centre of enlargement and the scale factor.## The Centre of Enlargement

The centre of enlargement is the point about which a shape is enlarged.When there is a negative scale factor, the centre of enlargement is between the original shape and the enlarged shape. The centre of enlargement can be described using Cartesian coordinates.## The Scale Factor

The scale factor describes how much larger (or smaller) the enlarged shape is compared to the original shape.The scale factor is found by dividing a length of a side of the enlarged shape to the length of the corresponding side of the original shape. When there is a negative scale factor, the enlarged shape is turned upside down. Care needs to be taken when finding the corresponding lengths.## How to Describe an Enlargement with a Negative Scale Factor

Describing an enlargement with a negative scale factor is easy.## Question

Describe the enlargement of the light blue shape to the dark blue shape below.## Step-by-Step:

# 1

Find a point on the shape.We will choose point

**A**on the shape.# 2

Find where the corresponding point on the enlarged shape will be.
When there is a negative scale factor, the enlarged shape looks like it has been turned upside down. In fact, the shape turned a half turn (rotated 180°).
Imagine turning the original shape a half turn.The corresponding point will be at the bottom right corner of the triangle.

# 3

Draw the corresponding point on the enlarged shape.
The corresponding point is the bottom right corner of the triangle.Point

**A'**is the corresponding point on the enlarged shape.# 4

Join the pair of corresponding points with a line.

# 5

Find another point on the shape.We will choose point

**B**on the shape.# 6

Find where the corresponding point on the enlarged shape will be.
When there is a negative scale factor, the enlarged shape looks like it has been turned upside down. In fact, the shape turned a half turn (rotated 180°).
Imagine turning the original shape a half turn.The corresponding point will be at the bottom left corner of the triangle.

# 7

Draw the corresponding point on the enlarged shape.
The corresponding point is the bottom left corner of the triangle.Point

**B'**is the corresponding point on the enlarged shape.# 8

Join the pair of corresponding points with a line.

# 9

Draw the centre of enlargement where the two lines meet.

# 10

Describe the centre of enlargement in Cartesian coordinates.
In our example, the Cartesian coordinates of the centre of enlargement is

**(3, 2)**.# 11

Find the length of a side on the enlarged shape.
The length of the side

**A'B'**is**4**.# 12

Find the length of the corresponding side on the original shape.
The length of the side

**AB**is**2**.# 13

Divide the length on the enlarged shape (4) by the length on the original shape (2).

$$\frac{A'B'}{AB} = \frac{4}{2}$$

$$\:\:\:\:\:\:\:\:\:\:\:\: = 2$$

# 14

Find the negative of this number (2) to find the scale factor.

$$Scale factor = -2$$

## Answer:

The shape has been enlarged about the centre of enlargement**(3, 2)**by a scale factor of

**−2**.