Bisector

Bisector

A line segment, line, or plane that divides a geometric figure into two congruent halves.

See also

Worked Example

Problem: Line segment AB has endpoints A(2, 4) and B(8, 10). Find the midpoint where a bisector of AB would cross the segment.

Step 1: Use the midpoint formula to find the point that divides AB into two equal parts.

M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)

Step 2: Substitute the coordinates of A(2, 4) and B(8, 10).

M = \left(\frac{2 + 8}{2},\; \frac{4 + 10}{2}\right) = \left(5,\; 7\right)

Step 3: Any line passing through M(5, 7) bisects segment AB because it splits the segment into two congruent parts, each of equal length. The perpendicular bisector is the special case that also crosses AB at a right angle.

Answer: The bisector crosses segment AB at the midpoint M(5, 7), dividing it into two segments each of length √((5−2)² + (7−4)²) = √(9 + 9) = 3√2.

Another Example

Problem: Angle PQR measures 70°. Ray QS is the angle bisector. What is the measure of each resulting angle?

Step 1: An angle bisector divides the angle into two congruent angles. Divide the original angle measure by 2.

\angle PQS = \angle SQR = \frac{70°}{2} = 35°

Step 2: Verify: the two smaller angles add back to the original angle.

35° + 35° = 70° \checkmark

Answer: Each resulting angle measures 35°.

Frequently Asked Questions

What is the difference between a bisector and a perpendicular bisector?

A bisector is any line, segment, or ray that divides a figure into two equal parts. A perpendicular bisector specifically bisects a line segment and also intersects it at a 90° angle. Every perpendicular bisector is a bisector, but not every bisector is perpendicular.

Can a bisector divide shapes other than segments and angles?

Yes. A bisector can divide areas, arcs, or even three-dimensional figures. For example, a diameter of a circle bisects the circle into two equal semicircles, and a plane can bisect a solid into two congruent halves.

Segment bisector vs. Angle bisector

A segment bisector passes through the midpoint of a line segment, splitting it into two segments of equal length. An angle bisector is a ray that starts at the vertex of an angle and divides it into two angles of equal measure. Both produce two congruent parts, but one acts on length and the other on angular measure.

Why It Matters

Bisectors appear throughout geometry proofs and constructions. The perpendicular bisector of a segment is the set of all points equidistant from its endpoints, which is the key idea behind circumscribing a circle around a triangle. Angle bisectors determine the incenter of a triangle, the center of the inscribed circle, making them essential in both theoretical and practical applications like engineering and design.

Common Mistakes

Mistake: Assuming any line through the midpoint of a segment is the perpendicular bisector.

Correction: A line through the midpoint bisects the segment, but it is only the perpendicular bisector if it also forms a 90° angle with the segment. There are infinitely many bisectors of a segment; only one is perpendicular.

Mistake: Confusing an angle bisector with a median or altitude of a triangle.

Correction: An angle bisector splits an angle into two equal parts. A median connects a vertex to the midpoint of the opposite side. An altitude drops perpendicularly from a vertex to the opposite side. These three lines generally point in different directions and meet the opposite side at different points.

Related Terms

Angle Bisector — Ray that bisects an angle into equal parts
Bisect — The verb meaning to divide into two equal parts
Line Segment — A common figure that a bisector divides
Midpoint — The point where a segment bisector crosses
Congruent — Describes the two equal parts a bisector creates
Perpendicular — Defines the special perpendicular bisector case
Plane — Can act as a bisector in three dimensions