CHAPTER 4: CONGRUENT TRIANGLES Section 4-7: Medians, Altitudes, and Perpendicular Bisectors

SYNOPSIS In section 4-7, we define 3 key concepts that apply to triangles. Having defined these 3 concepts, they will be used in 4 new theorems.

MEDIAN A median of a triangle is a segment that joins a vertex to the midpoint of its opposite side. Examples: Note that there are 3 different medians for the same triangle. X X X Y Z Y Z Y Z

ALTITUDE An altitude of a triangle is a segment that is perpendicular to the side opposite a vertex. Examples: Note that there are 3 different altitudes for the same triangle. X X X Z Y Z Y Z Y

ALTITUDE: RIGHT TRIANGLES The previous triangle was acute, so all altitudes were contained within the triangle. 2 of the 3 altitudes of right triangles are legs of the triangle per the images below: X X X Z Z Z Y Y Y

ALTITUDE: OBTUSE TRIANGLES Acute and right triangles have no exterior altitudes. Obtuse triangles have 2 exterior altitudes along with one interior altitude per the diagrams below: X X X Z Y Z Y Z Y

PERPENDICULAR BISECTOR A perpendicular bisector of a segment is a line, ray, or segment that intersects a segment at its midpoint and is perpendicular to the same segment. Line m is the ┴ bisector of segment AB as AE ≡ BE m A B E

THEOREM 4-5 THEOREM 4-5: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. D A B E

THEOREM 4-6 THEOREM 4-6: If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

THEOREM 4-7 THEOREM 4-7: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

THEOREM 4-8 THEOREM 4-8: If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle.

ALWAYS, SOMETIMES, OR NEVER An altitude is ______ perpendicular to the opposite side. A median is ______ perpendicular to the opposite side. An altitude is ______ an angle bisector. An angle bisector is ______ perpendicular to the opposite side. A perpendicular bisector of a segment is ______ equidistant from the endpoints of the segment. Always Sometimes

Suppose OG bisects TOY. What can you deduce if you also know that: Point J lies on OG? A point K is 13 cm from OT and 13 cm from OY? The distance from J to OT equals the distance from J to OY. K lies on OG

CLASSWORK/HOMEWORK Pg. 155, Classroom Exercises 1-6, 8-10 Pg. 156, Written Exercises 1-5, 7-14