Scale factor problems
In a nutshell
Scale factor problems involve similar shapes. They allow enlargements to be made whilst keeping the ratio of the sides the same.
Scale factors
A scale factor 'scales up' the ratio of the sides to one another, resulting in similar shapes.
Definitions
Scale factor
 When enlarging a shape, each side is multiplied by the same number. This number is known as the scale factor.

Similar shape  Shapes whose sides have the same ratio between them when simplified. They share the scale factor as a factor.

Working out lengths using a scale factor
Missing lengths can be worked out using a scale factor.
Example 1
The triangle below has sides of lengths $5cm, 7cm$ and $10cm$. What would the lengths be if this triangle was enlarged by a scale factor of two? Write your answer as a ratio.
Multiply each side by two.
$5cm \times 2 = {10cm} \\7cm \times 2 = {14cm} \\10cm \times 2 = {20cm}$
Write the lengths as a ratio.
$\underline{10cm:14cm:20cm}$
Working out the scale factor using lengths
You might also be asked to work out the scale factor. To do this, divide the end length by the starting length.
Note: Make sure that you are using the same side on each shape when picking your start and end length!
Example 2
A triangle has been enlarged. In terms of ratio, its sides have gone from $3cm:6cm:5cm$ to $12cm:24cm:20cm$. Work out the scale factor.
Pick a length and divide the final length by the start length.
$12cm\div3cm = 4cm$
Remove the units.
The scale factor is $\underline4$.