5-1 Bisectors of Triangles The student will be able to: 1. Identify and use perpendicular bisectors in triangles. 2. Identify and use angle bisectors in triangles.

Perpendicular Bisector Theorem Segment Bisector:Perpendicular Bisector: Perpendicular Bisector Theorem – A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment.

Example 1 & 2: Find each measure

3. Example 3 & 4:

You Try it: Find each measure

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.

Circumcenter Theorem – If you draw a perpendicular bisector from each side of a triangle, the 3 perpendicular bisectors intersect at a point called circumcenter that is equidistant from the vertices of the triangle. J is the circumcenter. The circumcenter will be: inside outside on

Examples 5, 6, & 7:

Example 8: A stove S, sink K, and refrigerator R are positioned in a kitchen as shown. Find the location for the center of an island work station so that it is the same distance from these three points.

You Try It: Two triangular-shaped gardens are shown below. Determine if a fountain can be placed at the circumcenter of each garden and still be inside the garden. Why or Why not?

Angle Bisectors An Angle Bisector is a special segment, ray, or line that divides an angle into two congruent angles.

Two properties of angle bisectors are: 1.A point is on the angle bisector of an angle if and only if it is equidistant from the sides of the angle. 2.The three angle bisectors of a triangle meet at a point, called the incenter of the triangle, that is equidistant from the three sides of the triangle. Point K is the incenter of ΔABC.

You Try It: If P is the incenter of ΔXYZ, find each measure.