LESSON 38Perpendicular and Angle Bisectors of Triangles

Vocabulary

An angle bisector of a triangle divides an angle into two congruent angles

The incenter of a triangle is the point of concurrency of angle bisectors

Vocabulary

The perpendicular bisector of a triangle divides a side into two congruent segments and is perpendicular to the side

The circumcenter of a triangle is the point of concurrency of perpendicular bisectors

Vocabulary

Inscribed circle is a circle that contains only one point from each side of the polygon (red circle)

Circumscribed circle is a circle that contains all vertices of a polygon (green circle)

Properties of Incenter and CircumcenterIncenter is equidistant from the sides Circumcenter is equidistant from the

vertices

Where is the circumcenter for a(n)…

Right Triangle Obtuse Triangle

Theorem 38-1: Triangle Angle Bisector TheoremIf a line bisects an angle of a triangle, then it divides the opposite side proportionally to the other two sides of the triangle.

Find AD, units are in feet

Application: City PlanningThree schools are located at points and If the city wants to build a playground equally distant from each school, where should it be located?

Since it needs to be equidistant, it should be at the circumcenter.

1. Draw ΔABC

2. Find midpoints

Application: City PlanningThree schools are located at points and If the city wants to build a playground equally distant from each school, where should it be located?

Since it needs to be equidistant, it should be at the circumcenter.

1. Draw ΔABC

2. Find midpoints

3. Draw perpendicular lines (only 2 are needed)

Application: City PlanningThree schools are located at points and If the city wants to build a playground equally distant from each school, where should it be located?

Since it needs to be equidistant, it should be at the circumcenter.

1. Draw ΔABC

2. Find midpoints

3. Draw perpendicular lines (only 2 are needed)

Review1. What is the intersection of medians?

Centroid

2. What is the intersection of altitudes?

Orthocenter

3. What is the intersection of angle bisectors?

Incenter

4. What is the intersection of perpendicular bisectors?

Circumcenter

Perpendicular and angles bisectors of triangles will prepare you for:

◦ Lesson 39: Inequalities in a Triangle

◦ Lesson 46: Triangle Similarity: AA, SSS, SAS

◦ Lesson 51: Properties of Isosceles and Equilateral Triangles

◦ Lesson 55: Triangle Midsegment Theorem