IB Maths AI 3.5 Notes

Perpendicular bisectors

A perpendicular bisector is a line that cuts a line segment into two equal parts at a right angle. This means it has two key properties:

• It passes through the midpoint of the segment.
• It is perpendicular to the segment.

To find the equation of a perpendicular bisector, we need two ideas from straight lines.

1. First, we need the midpoint of the segment. 
2. Second, we need the gradient of the perpendicular line. If a line has gradient $m$, then a perpendicular line has gradient $-\frac{1}{m}$.

There are two special cases to be aware of:

• If the original line segment is horizontal, its gradient is $0$. The perpendicular bisector will be a vertical line.
• If the original line segment is vertical, its gradient is undefined. The perpendicular bisector will be a horizontal line.

For example, if a segment has endpoints $(1,4)$ and $(5,4)$, it is horizontal. Its midpoint is $(3,4)$, so the perpendicular bisector is the vertical line $x=3$.