Geometry CP, Nov 26 Bisectors of Triangles Book Section: 5-1 Essential Question: What do you get when you bisect sides and angles of triangles? Standards: CCSS G.CO.10, G.MG.3 Perpendicular Bisector • A bisector is any segment, line, or plane that intersects a segment at its midpoint. • A perpendicular bisector is a bisector that is also perpendicular to the segment. Examples Perpendicular Bisector Theorems Example 1 Use the Perpendicular Bisector Theorems A. Find BC. Example 1 Use the Perpendicular Bisector Theorems B. Find XY. More Definitions • Concurrent Lines – Three or more lines that intersect at a common point. • Point of Concurrency – The point at which concurrent lines intersect. • Circumcenter – The point of concurrency of the perpendicular bisectors of the sides of a triangle Circumcenter Theorem Oh where, oh where can he be? • The circumcenter can be on the interior, exterior, or side of a triangle. Example 2 Use the Circumcenter Theorem GARDEN A triangular-shaped garden is shown. Can a fountain be placed at the circumcenter and still be inside the garden? Example 2 Use the Circumcenter Theorem Copy ΔXYZ, and use a ruler and protractor to draw the perpendicular bisectors. The location for the fountain is C, the circumcenter of ΔXYZ, which lies in the exterior of the triangle. C Answer: No, the circumcenter of an obtuse triangle is in the exterior of the triangle. Another Definition • Angle bisector – A line or plane that divides an angle into two equal parts. Angle Bisector Theorems Example 3 Use the Angle Bisector Theorems A. Find DB. Example 3 Use the Angle Bisector Theorems B. Find mWYZ. Examples Last Definition Today • Incenter (of a triangle) – The point of concurrency of the angle bisectors of a triangle. Incenter Theorem Example 4 Use the Incenter Theorem A. Find ST if S is the incenter of ΔMNP. Example 4 Use the Incenter Theorem B. Find mSPU if S is the incenter of ΔMNP. Examples Classwork: Textbook p.329, 9-13; p.330, 21, 22, 23, 24, 27-30 Homework: HW Due 11/28, 1- .