Perpendicular Bisectors

Video lesson · from 32:021 worked example

One lesson video covers all of The Straight Line, so it opens at 32:02 for this topic — not from the beginning.

At right angles

A line which divides another line into two equal parts.

i.e. a line which cuts through the midpoint of another line.

A line which divides another line into two equal parts at right angles.

i.e. a line which cuts through the midpoint of another line at right angles.

The three perpendicular bisectors of a triangle meet at the circumcentre.

The circumcentre is the centre of the triangle's circumcircle which passes through all vertices of the triangle.

Two steps, both needed: the midpoint of the line and the perpendicular gradient. Missing either one is the usual way marks are lost.
Perpendicular means negative reciprocal: flip the fraction and change the sign — doing only one of the two is the classic error.
Use the midpoint in the equation: $y - b = m(x - a)$ takes the midpoint, not either original endpoint.
Set the working out in order: midpoint, then original gradient, then perpendicular gradient, then equation. Each line should follow from the one above.

Example 1

A is the point $(-2, 1)$ and B is the point $(4, 7)$ . Find the equation of the perpendicular bisector of AB.

1. Midpoint of AB:

\begin{aligned} M &= \left(\frac{-2+4}{2}, \frac{1+7}{2}\right) \\ &= (1, 4) \end{aligned}

2. Gradient of AB:

\begin{aligned} m_{AB} &= \frac{7 - 1}{4 - (-2)} \\ &= \frac{6}{6} \\ &= 1 \end{aligned}

3. Perpendicular Gradient:

m_{\perp} = -1

4. Equation:

y - 4 = -1(x - 1)

y - 4 = -x + 1

y = -x + 5 \text{ or } x + y - 5 = 0