What Are Enlargements?

An enlargement is a type of transformation that changes the size of a shape but keeps its proportions (angles and relative side lengths) exactly the same. The result is a shape that is similar to the original — it looks the same, just bigger or smaller.

Enlargement

A transformation that produces an image that is the same shape as the original but a different size, using a centre of enlargement and a scale factor.

Every enlargement needs two pieces of information:

A centre of enlargement — the fixed point from which the shape expands or shrinks
A scale factor — the number that tells you how many times larger (or smaller) the image is compared to the original

Analogy

Think of enlargement like projecting an image onto a wall with a torch. The torch is the centre of enlargement, and moving it closer or farther from the wall changes the scale factor — the image gets bigger or smaller, but it always keeps the same shape.

Scale Factor

The scale factor ( $k$ ) tells you how the size of the shape changes during an enlargement.

$k = \frac{length on image}{corresponding length on original}$

If $k > 1$ : the image is larger than the original (a true enlargement)
If $0 < k < 1$ : the image is smaller than the original (a reduction, sometimes called a diminishment)
If $k = 1$ : the image is the same size as the original (congruent)
If $k < 0$ : the image is on the opposite side of the centre of enlargement (inverted)

Example

A triangle has sides of length 3 cm, 4 cm, and 5 cm. It is enlarged by a scale factor of $k = 2$ .

New side lengths:

$3 \times 2 = 6$ cm
$4 \times 2 = 8$ cm
$5 \times 2 = 10$ cm

The triangle is now twice as large, but all angles remain the same.

Warning

A common mistake is thinking that the area also doubles when the scale factor is 2. In fact, the area is multiplied by $k^{2}$ . So if $k = 2$ , the new area is $2^{2} = 4$ times the original area!

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