Midpoints & Segment Bisectors

The midpoint of a segment is the point exactly halfway between its endpoints. It splits the segment into two congruent pieces: if $M$ is the midpoint of $\overline{AB}$ , then $\displaystyle AM = MB = \frac{1}{2}AB$ . Those three equal signs are the whole lesson.

A segment bisector is any line, segment, ray, or plane that passes through the midpoint. The bisector itself can cross at any angle — what matters is that it cuts the segment into two equal halves. Problems in this lesson hand you one of the equal pieces, or the whole, or two expressions for the halves, and ask for what is missing.

What the midpoint gives you

If $M$ is the midpoint of $\overline{AB}$ , you get three facts at once: $AM = MB$ , $\displaystyle AM = \frac{1}{2}AB$ , and $AB = 2 \cdot AM$ . Every midpoint problem uses one of these.

Given the whole, halve it to find each piece. Given one piece, double it to find the whole. The most common error is halving when you should double — decide first whether the given length is a half or the whole. The figure below shows $M$ splitting $\overline{AB}$ into two congruent halves.

Segment bisectors

A segment bisector is anything that passes through the midpoint of the segment: a line, a ray, another segment, even a plane. The only thing a bisector guarantees is $AM = MB$ .

In particular, a bisector does not have to be perpendicular. A perpendicular bisector is the special case that both passes through the midpoint and forms a right angle — if a problem does not say perpendicular, do not assume a right angle, even if the picture happens to look like one.

Solving with algebra

When the two halves are expressions like $AM = 4x - 3$ and $MB = 2x + 9$ , the midpoint gives the equation: set the halves equal and solve for $x$ .

Then reread the question. If it asks for $x$ , you are done. If it asks for a piece, substitute $x$ into that expression. If it asks for the whole segment, find one piece and double it.

Worked examples

Example 1: from the whole to a half

$M$ is the midpoint of $\overline{AB}$ and $AB = 26$ . Find $AM$ .

The midpoint makes each piece half of the whole

AM = \dfrac{1}{2}AB

Substitute

AM = \dfrac{1}{2}(26)

Simplify

AM = 13

Answer: $AM = 13$

Example 2: from a half to the whole

$M$ is the midpoint of $\overline{GH}$ and $GM = 9$ . Find $GH$ .

The two halves are congruent

GM = MH = 9

Add the halves

GH = 9 + 9

Simplify

GH = 18

Answer: $GH = 18$

Example 3: halves written with a variable

$M$ is the midpoint of $\overline{AB}$ , with $AM = 3x - 1$ and $MB = 2x + 4$ . Find $x$ , then $AB$ .

The midpoint makes the halves equal

3x - 1 = 2x + 4

Subtract

2x

from both sides

x - 1 = 4

Add

1

to both sides

x = 5

Find one half

AM = 3(5) - 1 = 14

Double it for the whole segment

AB = 2(14) = 28

Answer: $x = 5$ and $AB = 28$

Try one yourself

Common questions

Does a segment bisector have to be perpendicular?

No. A bisector only has to pass through the midpoint — it can cross at any angle. Perpendicular bisector is a stronger condition: through the midpoint and at a right angle. Only use the right angle when the problem says perpendicular or marks it in the figure.

What is the difference between a midpoint and a bisector?

The midpoint is a point — the exact halfway spot on the segment. A bisector is a figure (line, ray, segment, or plane) that passes through that point. The bisector cuts the segment in half precisely because it goes through the midpoint.

How do I know whether to halve or double?

Ask what the given length is. If you are given the whole segment and need a piece, halve. If you are given a piece and need the whole, double. Sketching the segment with $M$ in the middle and labeling the given length makes the choice obvious.

How is this related to the midpoint formula?

Same idea, different setting. On the coordinate plane, the midpoint formula $\displaystyle M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ finds the coordinates of the halfway point. This lesson works with lengths instead of coordinates, but both rest on the same fact: the midpoint splits the segment into congruent halves.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1