1. Medians, Altitudes and Perpendicular Bisectors
Section 4-7
Medians, Altitudes and Perpendicular Bisectors

2. Median
connects the vertex to the midpoint of the opposite side

3. Thus, every triangle has three medians.

4. Altitude
the perpendicular segment from a vertex to a line that contains the opposite side B F D A C E

5. For obtuse triangles, two altitudes fall outside the figure.
You extend the base of the triangle so that it intersects the altitude at a right angle. B B C A C H A J

6. For obtuse triangles, there is still one altitude in the triangle.K C A

7. In Right Triangles – Two of the altitudes lie on the legs of the triangle. The 3rd is inside.
is an altitude
is an altitude
is an altitude

8. Perpendicular bisector of a segment
is a line (or ray or segment) that is perpendicular to the bisector at its midpoint l J K

9. In a given plane, there is exactly one perpendicular to a segment at its midpointK J

10. 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 A l B C

11. 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. A B C X

12. Theorem 4-7
If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. B C A X Y P Z

13. Theorem 4-8
If a point is equidistant from the sides of an angle, then the point lies on the bisector of the angle. B C A P

14. 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?
Sometimes
Sometimes

15. Always, Sometimes or Never?
An angle bisector is ___________ perpendicular to the opposite side.
A perpendicular bisector is ________ perpendicular to a segment at its midpoint.
A perpendicular bisector of a segment is __________ equidistant from the endpoints of the segment.
Always
Always