Enlarging a Shape with a Negative Scale Factor
(KS3, Year 8)

Enlarging a Shape with a Negative Scale Factor

A shape can be enlarged with a negative scale factor. If the scale factor is negative, the shape is enlarged on the other side of the centre of enlargement and it is turned upside down.

scale factor negative

How to Enlarge a Shape with a Negative Scale Factor

Enlarging a shape with a negative scale factor is easy.

Question

Enlarge the shape below by a scale factor of −2 about the centre of enlargement (5, 5).

Step-by-Step:

1

Plot the centre of enlargement. In our example, the Cartesian coordinates of the centre of enlargement is (5, 5). It is 5 units along the x-axis and 5 units up the y-axis. enlarge a shape scale factor negative step 1

enlarge a shape scale factor negative step 1

Each point on the shape is enlarged by the same amount. Let us choose a point on the shape and transform it. We will transform point A.

2

Draw a line from point A to the centre of enlargement. enlarge a shape scale factor negative step 2

Note: It is useful to extend the line beyond the centre of enlargement.

3

Measure the length of the line from the point to the centre of enlargement. In our example, the point is 1 diagonal unit from the centre of enlargement (or 1 unit across and 1 up). enlarge a shape scale factor negative step 3

enlarge a shape scale factor negative step 3

4

Multiply this distance (2) by the scale factor. Ignore the − sign in front of the scale factor for now.

Scaled distance = Distance × Scale factor

Scaled distance = 1 diagonal units × 2

Scaled distance = 2 diagonal units

The distance to the transformed point is 2 diagonal units. Now we use the − sign in the scale factor. Because the scale factor is negative, the distance will be on the other side of the centre of enlargement. The 2 diagonal units will be 2 units to the left and 2 units down.

5

Measure the distance found in Step 4 along the line drawn in Step 2. Measure it from the centre of enlargement going away from the shape. This is the point on the enlarged shape, which we will call A'. enlarge a shape scale factor negative step 5

enlarge a shape scale factor negative step 5

We have transformed point A to point A' on the enlarged shape. Repeat for points B and C. enlarge_a shape scale factor negative example A B C

Answer:

With all the vertices (corners) of the shape transformed, the enlarged shape can be drawn:

By multiplying the shape by a scale factor of −2, the enlarged shape is 2 times larger and 2 times the distance from the centre of enlargement. It is also the other side of the centre of enlargement and turned upside down.

Lesson Slides

The slider below shows another real example of how to enlarge a shape with a negative scale factor.

This page was written by Stephen Clarke.

What Is an Enlargement?

An enlargement resizes a shape. An enlargement makes a shape larger or smaller. An enlargement is a type of transformation.

Negative, Fractional Scale Factors

A scale factor can be negative and a fraction. For example, a scale factor of −¹⁄₂ will also enlarge a shape on the other side of the center of enlargement and turned upside down. $negative_fraction_scale_factor$ The shape is on the other side of the center of enlargement and turned upside down (because of the − sign) but it is also smaller (because of the fraction).

Enlarging a Shape with a Negative Scale Factor(KS3, Year 8)