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

Median connects the vertex to the midpoint of the opposite side

Thus, every triangle has three medians.

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

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

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

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

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

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

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

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

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

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

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

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