What is a Perpendicular Bisector?

Perpendicular Bisector: A line that passes through the midpoint of a line segment and is perpendicular (at 90°) to it. Every point on a perpendicular bisector is equidistant from the two endpoints of the original segment.

Perpendicular bisectors combine two key ideas from coordinate geometry:

Midpoints , finding the exact centre of a line segment
Perpendicular slopes , using the negative reciprocal relationship between slopes

They appear in problems involving symmetry, equidistant points, and triangle geometry (circumcentres).

Note

IB Math AI SL context: In this course you are expected to use your GDC to check your working , for example, graphing two lines to confirm their intersection point, or using the GDC's equation solver. Always show algebraic working first, then use the GDC to verify.

The Midpoint Formula

The first step in finding a perpendicular bisector is always to locate the midpoint of the given line segment.

Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the midpoint $M$ is:

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

This is simply the average of the $x$ -coordinates and the average of the $y$ -coordinates.

Example

Find the midpoint of the segment joining $A (2, 3)$ and $B (6, 7)$ .

$M = (\frac{2 + 6}{2}, \frac{3 + 7}{2}) = (4, 5)$

The midpoint is $(4, 5)$ .

Exam Tip

The midpoint formula is given in the IB formula booklet, but it is so frequently used that you should know it instantly without looking it up.

Free preview

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What is a Perpendicular Bisector?

Perpendicular bisectors combine two key ideas from coordinate geometry:

Midpoints , finding the exact centre of a line segment
Perpendicular slopes , using the negative reciprocal relationship between slopes

They appear in problems involving symmetry, equidistant points, and triangle geometry (circumcentres).

Note

The Midpoint Formula

The first step in finding a perpendicular bisector is always to locate the midpoint of the given line segment.

Given two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , the midpoint $M$ is:

$M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

This is simply the average of the $x$ -coordinates and the average of the $y$ -coordinates.

Example

Find the midpoint of the segment joining $A (2, 3)$ and $B (6, 7)$ .

$M = (\frac{2 + 6}{2}, \frac{3 + 7}{2}) = (4, 5)$

The midpoint is $(4, 5)$ .

Exam Tip

The midpoint formula is given in the IB formula booklet, but it is so frequently used that you should know it instantly without looking it up.

Free preview

SL 3.5—Intersection of lines, equations of perpendicular bisectors

What is a Perpendicular Bisector?

The Midpoint Formula

10 more sections in this topic

SL 3.5—Intersection of lines, equations of perpendicular bisectors

What is a Perpendicular Bisector?

The Midpoint Formula

10 more sections in this topic