Enlargements with a centre and the modulus of a complex number

Enlargement with centre the origin

An enlargement changes the size of a shape but not its shape. It is described by two things:

• the centre of enlargement, a fixed point that does not move;

• the scale factor $k$.

Each point $P$ is sent to an image point $P'$ on the ray from the centre $O$ through $P$, so that $|OP'| = k\,|OP|$.

• If $k > 1$, the image is bigger (further from the centre).

• If $0 < k < 1$, the image is smaller (closer to the centre).

When the centre is the origin, the rule is very simple: multiply both coordinates by $k$.

Why? The point $(x, y)$ is $x$ units across and $y$ units up from the origin. Making the distance $k$ times as big makes both of these $k$ times as big.

Enlargement with any centre

If the centre $C$ is not the origin, the rule "multiply the coordinates" no longer works directly. Instead, work with the displacement from the centre:

1. Find the vector from $C$ to $P$: subtract the centre's coordinates from $P$'s.

2. Multiply it by $k$.

3. Add it back to the centre.

This is the same idea as before: the origin case is just $c_1 = c_2 = 0$.

Finding the centre

If you know a point and its image, the centre lies on the line through them. You can use the formula and solve for the centre.

Effect on length and area

Under an enlargement with scale factor $k$:

Why $k^2$? A rectangle $2 \times 3$ has area $6$. With $k = 2$ it becomes $4 \times 6 = 24 = 6 \times 2^2$. Both length and width are multiplied by $k$, so the area is multiplied by $k \times k$.

The modulus of a complex number

A complex number $z = a + ib$ is shown on the Argand diagram as the point $(a, b)$: the real part $a$ along the horizontal axis and the imaginary part $b$ up the vertical axis.

The modulus $|z|$ is the distance from the origin to that point. By the theorem of Pythagoras:

To add complex numbers, add the real parts and the imaginary parts separately, then find the modulus of the result.

Link with enlargements

All complex numbers with the same modulus $r$ are at distance $r$ from the origin, so they lie on the circle $x^2 + y^2 = r^2$. Multiplying $z$ by a real number $k > 0$ multiplies its modulus by $k$: $|kz| = k|z|$. On the Argand diagram, this is an enlargement with centre the origin and scale factor $k$.

Back to the start: when you zoom a photo, each point moves along a ray from the centre of the zoom and its distance from the centre is multiplied by $k$. That is exactly an enlargement.

Summary

• An enlargement has a centre and a scale factor $k$. The image $P'$ lies on the ray from the centre through $P$, with $|CP'| = k|CP|$.

• Centre the origin: $(x, y) \to (kx, ky)$. Any centre $C$: subtract $C$, multiply by $k$, add $C$ back.

• $0 < k < 1$ makes the shape smaller; $k > 1$ makes it bigger.

• Lengths and perimeters are multiplied by $k$; areas by $k^2$; angles stay the same.

• The modulus $|a + ib| = \sqrt{a^2 + b^2}$ is the distance from the origin on the Argand diagram, and $|kz| = k|z|$.