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

2. 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.

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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.

10. 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.

11. 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.

12. 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

13. 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

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