Perpendicular and Angle Bisectors

G.CO.9 Prove theorems about lines and angles... points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Goals for this section:

Students should be able to understand the perpendicular bisector theorem and the angle bisector theorem and how it could be applied to real-life situations.

Perpendicular Bisector Theorem

Theorem: If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

Let's break it down:

  • Notice the perpendicular bisector forms a right angle (perpendicular) with segment AB.
  • Notice the perpendicular bisector splits segment AB into two equal halves (bisects).
  • If you pick any point on the perpendicular bisector, A and B will be the same distance away from P (equidistant).
      • That is AP ≅ BP.

Extra Property!

  • The angles formed by MAP and MBP are congruent!
  • The triangle formed by the dotted lines (APB) is called an isosceles triangle.

Angle Bisector Theorem

Theorem: If a point is on an angle bisector, then the point is equidistant from the sides of the angle.

Let's break it down:

  • An angle bisector divides an angle in two equal halves.
  • If you pick any point P on the angle bisector, it will be the same distance away from the sides of the angle (equidistant) when you draw a perpendicular line to the sides.

External Resources

Perpendicular Bisector Theorem

Angle Bisector Theorem

Keywords: perpendicular bisector theorem, angle bisector theorem, bisect, perpendicular