Similar figures
Look at these similar figures:
B is an enlargement of A. The lengths have doubled, but the angles have stayed the same.
Remember: For any pair of similar figures, corresponding sides are in the same ratio, and corresponding angles are equal.
Linear scale factor
The size of an enlargement/reduction is described by its scale factor.
For example, a The ratio of corresponding lengths in similar shapes, ie how much larger or smaller the shapes are. 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.
To calculate the Scale Factor, we use the following:
\(S{F_{Enlargement}} = \frac{{Big}}{{Small}}\)
\(S{F_{{\mathop{\rm Re}\nolimits} duction}} = \frac{{Small}}{{Big}}\)
You can get the 'big' and 'small' from the corresponding sides on the figures.
Example
The rectangles pqrs and PQRS are similar. What is the length of PS?
Answer
PS is on the bigger rectangle, therefore we will be calculating an enlargement scale factor first.
\(S{F_{Enlargement}} = \frac{{Big}}{{Small}} = \frac{7}{4}\)
Therefore rectangle PQRS is \(\frac{7}{4}\) times bigger than rectangle pqrs.
So, \(PS = \frac{7}{4} \times 9 = 15.75cm\)
(You can type in your calculator 7 ÷ 4 × 9)
Question
wxyz and WXYZ are similar figures. What is the length of XY?
XY is on the bigger figure, therefore we will be calculating the enlargement scale factor.
\(S{F_{Enlargement}} = \frac{{Big}}{{Small}} = \frac{9}{8}\)
Therefore figure WXYZ is \(\frac{9}{8}\) times bigger than figure wxyz.
So, \(XY = \frac{9}{8} \times 4 = 4.5cm\)
Question
What is the size of angle WXY?
Remember that the angles in similar figures stay the same. So angle WXY is 57°.
Question
The sides of a rectangle measure 8cm and 6cm.
If the rectangle is to be enlarged using scale factor \(\frac{3}{2}\) what will be the new lengths of the sides?
\(\frac{3}{2} \times 8 = 12cm\)
\(\frac{3}{2} \times 6 = 9cm\)
The new lengths will be 12cm and 9cm
Question
The sides of a rectangle measure 20cm and 28cm.
If the rectangle is to be enlarged using scale factor \(\frac{3}{4}\) what will be the new lengths of the sides?
\(\frac{3}{4} \times 20 = 15cm\)
\(\frac{3}{4} \times 28 = 21cm\)
The new lengths will be 15cm and 21cm
