Perpendicular Bisector
Overview
A perpendicular bisector does two jobs at once: it is perpendicular to a segment and it passes through the segment's midpoint. Its key property is about distance — every point on the perpendicular bisector is the same distance from the two endpoints of the segment it bisects. In a triangle, the three perpendicular bisectors meet at the circumcenter, which is therefore the same distance from all three vertices.
Vocabulary
- Perpendicular Bisector
- A segment, ray, or line that is perpendicular to a segment at its midpoint.
- Equidistant
- The same distance from two or more points or objects.
- Concurrent
- Three or more lines or segments that intersect at a single point.
- Circumcenter
- The point where the perpendicular bisectors of the three sides of a triangle meet; it is equidistant from the three vertices of the triangle.
Main Ideas
- Any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints.
- A perpendicular bisector does not have to come from a vertex of the triangle — it is built on a side, at that side's midpoint.
- A triangle has up to three perpendicular bisectors, one for each side, and they are concurrent at the circumcenter.
- The circumcenter is equidistant from the vertices of the triangle; it lies inside an acute triangle, outside an obtuse triangle, and on the hypotenuse of a right triangle.
- Because the bisector passes through the midpoint, the two halves of the bisected segment are equal — a second equation alongside the equidistant property.
Lesson Notes
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Lesson Video
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Practice Problems
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