Perpendicular and Angle Bisectors of a Triangle Sec 5.2 Goal: To use properties of perpendicular bisectors of a triangle. To use properties of angle bisectors of a triangle.

Perpendicular Bisectors of an Obtuse Triangle Review: The perpendicular bisector of a triangle is a line that is perpendicular to a side of the triangle at the midpoint of the side. Point of concurrency or circumcenter The three perpendicular bisectors are outside the triangle for an obtuse triangle.

Obtuse Triangle The perpendicular bisectors of a triangle intersect at a point P that is equidistant from the vertices of the triangle. Theorem 5.5 Point of concurrency or circumcenter PA = PB = PC

Perpendicular Bisector of an Acute Triangle Point of concurrency or circumcenter The three perpendicular bisectors are inside the triangle for an acute triangle.

Acute Triangle Point of concurrency or circumcenter PA = PB = PC The perpendicular bisectors of a triangle intersect at a point P that is equidistant from the vertices of the triangle. Theorem 5.5

Perpendicular Bisector of a Right Triangle Point of concurrency or circumcenter The three perpendicular bisectors are on the triangle for a right triangle.

Right Triangle Point of concurrency or circumcenter PA = PB = PC The perpendicular bisectors of a triangle intersect at a point P that is equidistant from the vertices of the triangle. Theorem 5.5

Angle Bisector of a Triangle Angle bisector An angle bisector of a triangle – a line that bisects an angle of the triangle.

Angle Bisectors of a Triangle The three bisectors are concurrent at a point P called the incenter

Angle Bisectors of a Triangle The three bisectors are concurrent at a point P called the incenter The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. Theorem 5.6 PD = PE = PF

Example The perpendicular bisectors met at point P. Find PC. Find DP.

Example The angle bisectors met at point M. Find MK. Find ZK.

Example You have three salesmen who are selling cars and you are looking for a location to build a plant to manufacture cars.