Fractional scale factor
You already know that the size of an enlargement is described by its scale factor.
For example, a scale factor of 2 means that the side-lengths of the new shape are twice the side-lengths of the original. A scale factor of 3 means that the side-lengths of the new shape are three times the side-lengths of the original.
Therefore, a scale factor of \(\frac{1}{2}\) means that the side-lengths of the new shape are half the side-lengths of the original.
When the scale factor is fractional and the shape decreases in size, we call this a reduction.
Example
The slideshow below goes through the steps to reduce a triangle using a scale factor of \(\frac{1}{2}\) and centre of enlargement O:
Image caption, Reducing a triangle with a scale factor of ½
A right-angled triangle.
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Question
What is the scale factor in this diagram?
The scale factor is \(\frac{1}{3}\)
Notice that OA is \({6}\) units and OA' is \({2}\) units.
BC is \({3}\) units and B'C' is \({1}\) unit, so each side on the image is \(\frac{1}{3}\) of the length of the original shape.
Always check which shape is the object and which shape is the image, to avoid confusing the scale factors.
For example, a scale factor of \({2}\) might be mistaken for a scale factor of \(\frac{1}{2}\), or a scale factor of \({3}\) mistaken for a scale factor of \(\frac{1}{3}\) if you start with the image and not the original object.
