How to Describe an Enlargement with a Negative Scale Factor

Describing an Enlargement with a Negative Scale Factor

A shape can be enlarged with a negative scale factor.

To describe an enlargement, we need to describe the center of enlargement and the scale factor.

The Center of Enlargement

The center of enlargement is the point about which a shape is enlarged.

When there is a negative scale factor, the center of enlargement is between the original shape and the enlarged shape.

The center 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 center of enlargement where the two lines meet.

10

Describe the center of enlargement in Cartesian coordinates.

In our example, the Cartesian coordinates of the center of enlargement is (3, 2).

We have found the center 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.

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 center of enlargement (3, 2) by a scale factor of −2.

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See Also

What is geometry? What are transformations? What is a rotation? Common rotations What are Cartesian coordinates?