4-7 Median, Altitude, and Perpendicular bisectors

Medians, Altitudes and Perpendicular Bisectors A median of a triangle is a segment from a vertex to the midpoint of the opposite side. D A B C

Medians, Altitudes and Perpendicular Bisectors Every triangle has 3 medians. The 3 medians of ABC are shown below. D B C A

Medians, Altitudes and Perpendicular Bisectors An altitude of a triangle is the perpendicular segment from a vertex to the line that contains the opposite side B C A

Medians, Altitudes and Perpendicular Bisectors In an acute triangle, the three altitudes are all inside the triangle. D B C A

Medians, Altitudes and Perpendicular Bisectors In a right triangle, two of the altitudes are parts of the triangle. They are the legs of the right triangle. The third altitude is inside the triangle. A B C D

Medians, Altitudes and Perpendicular Bisectors In an obtuse triangle, two of the altitudes are outside the triangle. ABC has two altitudes outside the triangle. A C B

Medians, Altitudes and Perpendicular Bisectors A perpendicular bisector of a segment is a line that is perpendicular to the segment at its midpoint. In the figure below, line m is a perpendicular bisector of AB. m A B

Perpendicular Bisector Theorem 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. AB 8 C If CD is the perpendicular bisector of AB than AC = BC D

10 Perpendicular Bisector Converse Theorem 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 AB 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 B X P Y C A Z

Theorem 4-8 If a point is equidistance from the side of an angle, then the point lies on the bisector of an angle B X P Y C A Z