Resizing from a point
An enlargement changes the size of a shape but keeps it the same shape, so the image is similar to the original. Unlike a translation, an enlargement is anchored to a fixed point called the centre of enlargement. Think of a projector: the bulb is the centre, and moving the screen further away makes a bigger picture of the same slide.
- Enlargement
- A transformation that makes a shape larger or smaller by a scale factor, measured from a fixed centre of enlargement. The image is similar to the original.
Scale factor and centre
The scale factor tells you how many times bigger the image is. The centre of enlargement tells you where the enlargement is measured from. To find an image point, measure the distance from the centre to the original point, then multiply that distance by the scale factor.
When the centre is the origin \((0,0)\), the shortcut is neat: just multiply every coordinate by the scale factor.
Draw the rays
To enlarge by hand, draw a straight line (a ray) from the centre through each vertex, then step out along it by the scale factor. The image vertices always sit on those rays, which is a quick way to check your work.
A worked example
Enlarge triangle with vertices \((1,1)\), \((2,1)\) and \((1,3)\) by scale factor \(2\), centre the origin. Find the image coordinates.
Fractional scale factors
A scale factor does not have to be a whole number. A scale factor between 0 and 1, such as \(\tfrac{1}{2}\), makes the shape smaller. It is still called an enlargement in this topic, even though the picture shrinks. To use it from the origin, multiply each coordinate by the fraction.
For example, enlarging \((4,6)\) by scale factor \(\tfrac{1}{2}\) about the origin gives \((2,3)\), which is half as far from the centre.
Where this is assessed
Multiplying coordinates by the scale factor is Criterion A (Knowing and understanding). A full description names the transformation as an enlargement and states both the scale factor and the centre, which is Criterion C (Communicating). Missing the centre costs marks. Scale drawings and model problems bring in Criterion D (Applying maths in context).
Check yourself
1. Enlarge the point (3,5) by scale factor 3, centre the origin. Where does it go? +
From the origin, multiply each coordinate by 3: \((3 \times 3,\ 5 \times 3) = \mathbf{(9,15)}\).
2. Enlarge the point (8,-4) by scale factor 1/2, centre the origin. Where does it go? +
Multiply each coordinate by \(\tfrac{1}{2}\): \((8 \times \tfrac{1}{2},\ -4 \times \tfrac{1}{2}) = \mathbf{(4,-2)}\). The point is now half as far from the centre.
3. What does a scale factor between 0 and 1 do to a shape? +
It makes the shape smaller, while keeping it the same shape (similar). Each length is multiplied by the fraction, so the image is closer to the centre.
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.