Special Segments of Triangles Sections 5.2, 5.3, 5.4

Perpendicular bisector theorem A point is on the perpendicular bisector if and only if it is equidistant from the endpoints of the segment.

Angle Bisector Theorem A point is on the bisector of an angle if and only if it is equidistant from the two sides of the angle.

Medians of a triangle A median of a triangle is a segment from a vertex to the midpoint of the opposite side.

Altitudes of a triangle An altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.

Concurrency The point of intersection of the lines, rays, or segments is called the point of concurrency.

Points of concurrency The point of concurrency of the three perpendicular bisectors a triangle is called the circumcenter. The point of concurrency of the three angle bisectors of a triangle is called the incenter. The point of concurrency of the three medians of a triangle is called the centroid. The point of concurrency of the three altitudes of a triangle is called the orthocenter. The incenter and centroid will always be inside the triangle. The circumcenter and orthocenter can be inside, on, or outside the triangle.

What is special about the Circumcenter? The perpendicular bisectors of a triangle intersect at a point that is equisdistant from the vertices of the triangle. PA = PB = PC

What is special about the Incenter? The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF

What is special about the Centroid? The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

What is special about the Orthocenter? There is nothing special about the point of concurrency of the altitudes of a triangle.

Assignment Pg. 306 #3, 5, 11-17 odds Pg. 313 #3-23 odds Pg. 322 #3-7odds, 17-21 odds, 33, 35