Section 4 7 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 A D E C

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 H A B C A J C

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

In Right Triangles – Two of the altitudes lie on the legs of rd the triangle. The 3 is inside. 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 l J M K

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

Always, Sometimes or Never? Always • An altitude is _______ perpendicular to the opposite side? Sometimes • A median is _______ perpendicular to the opposite side? Sometimes • An altitude is ____ an angle bisector?

Always, Sometimes or Never? Sometimes • An angle bisector is ______ perpendicular to the opposite side. Always • A perpendicular bisector is ____ perpendicular to a segment at its midpoint. • A perpendicular bisector of a segment Always equidistant from the is _____ endpoints of the segment.