5.1 Notes Bisectors of Triangles

Perpendicular Bisectors We learned earlier that a segment bisector is any line, segment, or plane that intersects a segment at its midpoint. If a bisector is also perpendicular to the segment, it is called a perpendicular bisector.

Perpendicular Bisectors

Example 1 a) Find the length of BC.

Example 1 b) Find the length of XY.

Example 1 c) Find the length of PQ.

Extra Vocab When three or more lines intersect at a common point, the lines are called concurrent lines. The point where concurrent lines intersect is called the point of concurrency. A triangle has three sides, so it also has three perpendicular bisectors. These bisectors are concurrent lines. The point of concurrency of the perpendicular bisectors is called the circumcenter of the triangle.

Circumcenter Theorem

The circumcenter can be on the interior, exterior, or side of a triangle.

Example 2 A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden?

Angle Bisectors We learned earlier that an angle bisector divides an angle into two congruent angles. The angle bisector can be a line, segment, or ray.

Angle Bisectors

Example 3 a) Find the length of DB.

Example 3 b) Find m  WYZ.

Example 3 c) Find the length of QS.

The angle bisectors of a triangle are concurrent, and their point of concurrency is called the incenter of a triangle.

Example 4 a) Find ST if S is the incenter of ΔMNP. c) Find m  SPU if S is the incenter of ΔMNP.