Enlarging and reducing shapes

A scale factor can be used to enlarge or reduce a shape.

Shape A below is enlarged by scale factor \(2\) to give Shape B.

Diagram of 2 shapes of the same proportions but different sizes

Linear Scale Factor

The scale factor describes the size of an enlargement or reduction.

For example, a scale factor of \(2\) means that the new shape is twice the size of the original.

A scale factor of \(3\) means that the new shape is three times the size of the original.

A scale factor of \(\frac{1}{2}\) means that the new shape is half the size of the original.

To calculate the scale factor, we use the following:

\[SF_{Enlargement}=\frac{Big}{Small}\]

\[SF_{Reduction}=\frac{Small}{Big}\]

You can get the 'big' and 'small' from the corresponding sides on the figures.

Question

Rectangles \(PQRS\) is an enlargement of rectangle \(pqrs\). What is the length of \(PS\)?

Two rectangles, one measuring 4cm x 9cm, the other with a height of 7cm and unknown length

\(PS\) is on the bigger rectangle, therefore we will be using an enlargement scale factor.

\[SF_{Enlargement}= \frac{Big}{Small}=\frac{7}{4}\]

Therefore \(PS\) is \(\frac{7}{4}\) times \(ps\).

So, \(PS = \frac{7}{4} \times 9 = 15.75cm\)

(You can type in your calculator \(7 \div 4 \times 9\))