Bisectors of Triangles

1. Bisectors of Triangles Geometry (Holt 5-2) K. Santos

2. Concurrent & Point of Concurrency When 3 or more lines intersect at one point, then the lines are said to be concurrent. Point of concurrency is the point where they intersect.

3. Circumcenter of a Triangle Circumcenter of a triangle—is the point of concurrency where the 3 perpendicular bisectors are concurrent.

4. Property of Circumcenter Circumcenter is equidistant to the vertices of the triangle Point on perpendicular bisector is equidistant to the endpoints of the segment (vertices of triangle) perpendicular bisectors congruent segments (equidistant)

5. Example--Circumcenter G is the circumcenterof triangle ABC. Find GC. B 13.4 D G E 7.3 A F C Circumcenter----perpendicular bisectors Perpendicular bisectors—equidistant from endpoints so GB = GA = GC Thus GC = 13.4

6. Circumscribed and inscribed Circumscribed—circle outside the triangle (figure) only vertices touch the circle Inscribed—circle inside the triangle (figure) each side touches circle once shape never goes into the circle

7. Incenter of a Triangle Incenter of a triangle—is the point of concurrency where the 3 angle bisectors are concurrent

8. Property of Incenter Incenter—angle bisectors Incenter is equidistant to the sides of the triangle angle bisectors congruent segments equidistant from sides perpendicularly

9. Example--Incenter and are angle bisectors of LMN. Find each measure. M Q 5 P 50 20 L N a. The distance from P to MN = QP so MN = 5 b. m < PMN m< MLN = 2(50) = 100 m< LMN = 180 – (100 +20) = 60 but m<LMN is bisected so m<PMN = 30