Enlargements — Mathslive.ie

Section 1 of 2

Enlarging a Shape

The words

An enlargement has a centre of enlargement, an object (the start shape) and an image (the new shape). Ray lines from the centre through each corner set where the image goes.

Scale factor

$|A'B'| = k\,|AB|$, where $k$ is the scale factor. $k > 1$ makes it bigger; $k < 1$ makes it smaller.

Worked example — a doubled side

A shape $ABCD$ is enlarged by scale factor $2$. If $|AB| = 5$, find $|A'B'|$.

$|A'B'| = k\,|AB| = 2 \times 5$

$|A'B'| = 10$ (the sides double)

$|A'B'| = 10$

Section 2 of 2

Area of an Enlargement

Area uses k squared

$\text{Area of image} = k^2 \times (\text{area of object})$. The area grows by the square of the scale factor.

Worked example — area after enlarging

Rectangle $ABCD$ has area $16$. It is enlarged by scale factor $2$. Find the area of the image.

$\text{Area of image} = k^2 \times \text{area} = 2^2 \times 16$

$= 4 \times 16 = 64$

$64$

You try

A shape is enlarged so that $|A'B'| = 3\,|AB|$. If $|AB| = 4$, find $|A'B'|$.

$|A'B'| = 3 \times 4$.

$|A'B'| = 12$

You try

To construct an enlargement of $\triangle OAB$ from centre $O$ with $|A'B'| = 3|AB|$, how far along each ray line does each image point go, compared with the object point?

Scale factor $k = 3$.

Each image point is $3$ times as far from the centre $O$ as the matching object point ($k = 3$).

$3$ times as far

That’s Class 9.

Scale factor $k$, $|A'B'| = k|AB|$, and area $= k^2 \times$ area. Class 10: similar triangles.