Perpendicular & Angle Bisectors

Objectives Identify and use ┴ bisectors and  bisectors in ∆s

Perpendicular Bisector A ┴ bisector of a ∆ is a line, segment, or ray that passes through the midpoint of one of the sides of the ∆ at a 90° . Side AB perpendicular bisector PAB C

┴ Bisector Theorems Theorem 5.1 – Any point on the ┴ bisector of a segment is equidistant from the endpoints of the segment. Theorem 5.2 – Any point equidistant from the endpoints of a segment lies on the ┴ bisector of the segment.

┴ Bisector Theorems (continued) Basically, if CP is the perpendicular bisector of AB, then PA ≅ PB. Side AB perpendicular bisector PAB C

┴ Bisector Theorems (continued) Since there are three sides in a ∆, then there are three ┴ Bisectors in a ∆. These three ┴ bisectors in a ∆ intersect at a common point called the circumcenter.

┴ Bisector Theorems (continued) Theorem 5.3 (Circumcenter Theorem) The circumcenter of a ∆ is equidistant from the vertices of the ∆. Notice, a circumcenter of a ∆ is the center of the circle we would draw if we connected all of the vertices with a circle on the outside (circumscribe the ∆). circumcenter

Example What is the length of AB?

Angle Bisectors of ∆s Another special bisector which we have already studied is an  bisector. As we have learned, an  bisector divides an  into two ≅ parts. In a ∆, an  bisector divides one of the ∆s  s into two ≅  s. (i.e. if AD is an  bisector then  BAD ≅  CAD) B D C

Angle Bisectors of ∆s (continued) Theorem 5.4 (Angle Bisector Theorem) – Any point on an  bisector is equidistant from the sides of the . Theorem 5.5 (Converse of the Angle Bisector Theorem) – Any point equidistant from the sides of an  lies on the  bisector.

Angle Bisectors of ∆s (continued) As with ┴ bisectors, there are three  bisectors in any ∆. These three  bisectors intersect at a common point we call the incenter. incenter

Angle Bisectors of ∆s (continued) Theorem 5.6 (Incenter Theorem) The incenter of a ∆ is equidistant from each side of the ∆.

Example What is the length of RM?

Your Turn What is the length of FD?