Section 4-7

1. 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 C A 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. B K C A

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

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

9. In a given plane, there is exactly one perpendicular to a segment at its midpoint M l K J

10. Theorem 4-5 A B C l • If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment

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

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

14. Always, Sometimes or Never? Always • 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? Sometimes • 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