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Describing an Enlargement with a Negative Scale Factor

(KS3, Year 8)

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.enlargement_center_of_enlargement_negative_scale_factorWhen 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.enlargement_negative_scale_factorThe 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. describe enlargement negative scale factor example

Step-by-Step:

1

Find a point on the shape.describe_enlargement_negative_scale_factor_step_1We 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.describe_enlargement_negative_scale_factor_step_2The 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.describe_enlargement_negative_scale_factor_step_3Point A' is the corresponding point on the enlarged shape.

4

Join the pair of corresponding points with a line. describe_enlargement_negative_scale_factor_step_4

5

Find another point on the shape.describe_enlargement_negative_scale_factor_step_5We 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.describe_enlargement_negative_scale_factor_step_6The 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.describe_enlargement_negative_scale_factor_step_7Point B' is the corresponding point on the enlarged shape.

8

Join the pair of corresponding points with a line. describe_enlargement_negative_scale_factor_step_8

9

Draw the centre of enlargement where the two lines meet. describe_enlargement_negative_scale_factor_step_9

10

Describe the centre of enlargement in Cartesian coordinates. In our example, the Cartesian coordinates of the centre of enlargement is (3, 2).
We have found the centre of enlargement. We can find the scale factor.

11

Find the length of a side on the enlarged shape. The length of the side A'B' is 4. describe_enlargement_negative_scale_factor_step_11

12

Find the length of the corresponding side on the original shape. The length of the side AB is 2. describe_enlargement_negative_scale_factor_step_12

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.

Lesson Slides

The slider below shows another real example of how to describe an enlargement with a negative scale factor.

What Is an Enlargement?

An enlargement resizes a shape. An enlargement makes a shape larger or smaller. An enlargement is a type of transformation.
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This page was written by Stephen Clarke.

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