Perpendicular and Angle Bisectors of a Triangle Sec 5.2

1
Perpendicular and Angle Bisectors of a
TriangleSec 5.2

• Goal
• To use properties of perpendicular bisectors of
  a triangle.
• To use properties of angle bisectors of a
  triangle.

2
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.
3
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
4
Perpendicular Bisector of an Acute Triangle
Point of concurrency or circumcenter
The three perpendicular bisectors are inside the
triangle for an acute triangle.
5
Acute 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
6
Perpendicular Bisector of a Right Triangle
Point of concurrency or circumcenter
The three perpendicular bisectors are on the
triangle for a right triangle.
7
Right 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
8
Angle Bisector of a Triangle
An angle bisector of a triangle a line that
bisects an angle of the triangle.
Angle bisector
9
Angle Bisectors of a Triangle
The three bisectors are concurrent at a point P
called the incenter
10
Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a
point that is equidistant from the sides of the
triangle. Theorem 5.6
The three bisectors are concurrent at a point P
called the incenter
PD PE PF
11
Example

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

12
Example

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

13
Example

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