Perpendicular Bisector
In short: A perpendicular bisector of a segment is a line that crosses the segment at its midpoint at a right angle, so every point on it is equally far from both endpoints.
GeometryGeometryG-CO.C.9
What is Perpendicular Bisector?
A perpendicular bisector does two jobs at once: it bisects a segment, meeting it exactly at the midpoint, and it is perpendicular to the segment, crossing at \(90^{\circ}\).
Its key property is a distance test. A point lies on the line if and only if it is equidistant from the two endpoints. In coordinates, find the midpoint, then use a slope that is the negative reciprocal of the segment’s slope.
How to spot it
- A figure shows a small square at the crossing point and matching tick marks on the two halves of the segment.
- The wording asks for a point equally far from two locations, such as two towns, two wells or two vertices.
- A coordinate problem asks for the equation of the line that cuts a segment in half at a right angle.
- A triangle problem needs the circumcenter, where three such lines meet.
The method
- Find the midpoint of the segment by averaging the endpoint coordinates.
- Find the segment’s slope, then flip it and change its sign to get the perpendicular slope.
- Write the line through the midpoint with that slope, in point-slope form, and simplify.
- Test a point on the line: its distances to the two endpoints must come out equal.
Perpendicular Bisector: solved example
Problem
Find the equation of the perpendicular bisector of the segment from \(A(2, -1)\) to \(B(8, 3)\).
Solution
Midpoint: \(\left(\frac{2 + 8}{2}, \frac{-1 + 3}{2}\right) = (5, 1)\). Slope of \(AB\): \(\frac{3 – (-1)}{8 – 2} = \frac{4}{6} = \frac{2}{3}\), so the perpendicular slope is \(-\frac{3}{2}\).
Point-slope form: \(y – 1 = -\frac{3}{2}(x – 5)\), which simplifies to \(y = -\frac{3}{2}x + \frac{17}{2}\). Check with \(P(1, 7)\): \(PA^2 = 1 + 64 = 65\) and \(PB^2 = 49 + 16 = 65\), so \(P\) is equally far from both ends.
Try it
Question 1
A segment has slope \(\frac{5}{4}\). What slope does its perpendicular bisector have?
Show the answer
Flip and negate: \(-\frac{4}{5}\).
Question 2
Point \(X\) lies on the perpendicular bisector of \(\overline{JK}\), and \(XJ = 13\) cm. What is \(XK\)?
Show the answer
Points on the line are equidistant from both endpoints, so \(XK = 13\) cm.
Perpendicular Bisector vs. Angle Bisector
The first one belongs to a segment, halves its length and meets it at \(90^{\circ}\). An angle bisector belongs to a corner, halves a degree measure and need not be perpendicular to anything. Different objects, different halves: one splits a distance, the other splits an opening.
Keep going
Lessons
Worksheets and quizzes
- Geometric Constructions – Worksheet
- Geometric Constructions – Quiz
- Coordinate Geometry and Proofs – Worksheet
- Coordinate Geometry and Proofs – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry