Enlargement

What you are looking at

The blue shape is the object, the yellow dot is the centre of enlargement, and dashed guide lines run from the centre through each vertex to its image. The working computes every image point as centre $+ k \times$ (point $-$ centre), in exact fractions, and gives the lengths and the area: a side of the image is $|k|$ times the side of the object, and the area is $k^2$ times — so a scale factor of $3$ gives nine times the area, and $\dfrac{1}{2}$ gives a quarter.

The slider walks $k$ through $-3, -\dfrac{5}{2}, -2 \ldots -\dfrac{1}{3}, \dfrac{1}{3}, \dfrac{1}{2} \ldots 3$. Between $0$ and $1$ the image shrinks towards the centre; at $k = 1$ it sits on the object; above $1$ it grows away. A negative $k$ puts the image on the other side of the centre, turned through $180^\circ$ — and $k = -1$ is the same as a rotation of $180^\circ$ about the centre. Press Play and the image walks through the whole range.

Try this

1. 1Drag the centre onto a vertex of the object. That vertex is its own image — $k$ times a distance of zero is zero — and the shape grows out from that corner.
2. 2Set $k = \dfrac{1}{2}$ and then $k = -\dfrac{1}{2}$: the images are the same size, mirrored through the centre.
3. 3Slide to $k = -1$ and compare the object and image: a half turn about the centre, which is why an exam answer may describe it either way.
4. 4Switch to Quadrilateral and set $k = 3$: the working shows the image area is $9$ times the object area. Count the squares to check.

Mistakes students make

• Enlarging from the wrong point. Each image vertex lies on the line from the *centre* through the original vertex, not on a line from the origin — unless the centre happens to be $(0, 0)$.
• Describing an enlargement with the scale factor only. Full marks need three things: the word enlargement, the scale factor, and the centre.
• Scaling the area by $k$. Lengths scale by $|k|$, areas by $k^2$: a scale factor of $2$ makes the area four times as large, not twice.

In the exam

• To find the centre from a diagram, draw the lines through corresponding vertices ($A$ to $A'$, $B$ to $B'$): they meet at the centre. The scale factor is image length ÷ object length, negative if the image is on the other side.
• Negative scale factors are Extended only; fractional ones appear on both papers.

Where it is taught

• Cambridge IGCSE Mathematics 0580/0980 · 7.1 Transformations — enlargement (negative scale factors: Extended)