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Enlarging a Shape with a Fractional Scale Factor

(KS3, Year 8)

Enlarging a Shape with a Fractional Scale Factor

A shape can be enlarged with a scale factor. Usually, this makes the shape larger. If the scale factor is a fraction (between 0 and 1), the enlargement makes the shape smaller.

$scale factor fraction$

How to Enlarge a Shape with a Fractional Scale Factor

Enlarging a shape with a fractional scale factor is easy.

Question

Enlarge the shape below by a scale factor of ¹⁄₃ about the centre of enlargement (1, 1). $enlarge a shape scale factor fraction example$

Step-by-Step:

1

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

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 the centre of enlargement to point A. $enlarge a shape scale factor fraction step 2$ Note: It is useful to extend the line beyond the point.

3

Measure the length of this line. In our example, the point is 3 diagonal units from the centre of enlargement (or 3 units across and 3 up). $enlarge a shape scale factor fraction step 3$

4

Multiply this distance (3) by the scale factor (¹⁄₃).

Scaled distance = Distance × Scale factor

Scaled distance = 3 diagonal units × ¹⁄₃

Scaled distance = 1 diagonal unit

The distance to the transformed point is 1 diagonal unit (or 1 unit across and 1 up).

5

Measure the distance found in Step 4 along the line drawn in Step 2. This is the point on the enlarged shape, which we will call A'. $enlarge a shape scale factor fraction step 5$

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

Answer:

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

$enlarge a shape scale factor fraction answer$ By multiplying the shape by a scale factor of ¹⁄₃, the enlarged shape is a ¹⁄₃ of the size and a ¹⁄₃ the distance from the centre of enlargement.

Lesson Slides

The slider below shows another real example of how to enlarge a shape with a fractional 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.

Why Do Fractional Scale Factors Shrink a Shape?

A scale factor multiplies the length of each side of the shape that is enlarged. If a scale factor is a fraction between 0 and 1, then each side is going to get smaller. For example, multiplying a length by ¹⁄₂ will make each side half as long as it was before - which is shorter. scale_factor_half_mini

Fractional, Negative Scale Factors

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

Enlarging a Shape with a Fractional Scale Factor