Segment Bisector — Definition, Formula & Examples

A segment bisector is any line, ray, segment, or point that cuts a line segment into two equal halves. The point where it crosses the segment is called the midpoint.

A segment bisector of a line segment $\overline{AB}$ is a geometric figure (point, line, ray, or segment) that intersects $\overline{AB}$ at its midpoint $M$ , such that $AM = MB$ . When the bisector is also perpendicular to $\overline{AB}$ , it is called a perpendicular bisector.

Worked Example

Problem: Segment

\overline{AB}

has endpoints

A(2, 4)

and

B(10, 4)

. A segment bisector crosses

\overline{AB}

at its midpoint

M

. Find the coordinates of

M

and confirm that

AM = MB

Find the midpoint: Use the midpoint formula to find

M

M = \left(\frac{2+10}{2},\; \frac{4+4}{2}\right) = (6,\, 4)

Verify equal lengths: Calculate the distance from

A

M

and from

M

B

AM = 6 - 2 = 4, \quad MB = 10 - 6 = 4

Conclusion: Since

AM = MB = 4

, any line, ray, or segment passing through

M(6,4)

is a segment bisector of

\overline{AB}

Answer: The midpoint is

M(6, 4)

, and both halves have length 4, confirming the bisector divides

\overline{AB}

equally.

Why It Matters

Segment bisectors are essential in compass-and-straightedge constructions, such as finding the perpendicular bisector of a side of a triangle. They also appear in coordinate geometry proofs and in real-world tasks like finding the exact center of a beam or wall.

Common Mistakes

Mistake: Thinking a segment bisector must be perpendicular to the segment.

Correction: Any line through the midpoint bisects the segment, regardless of the angle at which it crosses. Only a perpendicular bisector crosses at a 90° angle.