Medians, Altitudes and Angle Bisectors

Every triangle has 1. 3 medians, 2. 3 angle bisectors and 3. 3 altitudes.

A B C Given  ABC, identify the opposite side 1. of A. 2. of B. 3. of C. BC AC AB

Any triangle has three medians. A B C L M N Let L, M and N be the midpoints of AB, BC and AC respectively. Hence, CL, AM and NB are medians of  ABC. Definition of a Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex of a triangle and a midpoint of the side opposite that vertex.

A B C D E F In the figure, AF, DB and EC are angle bisectors of  ABC. Definition of an Angle Bisector of a Triangle A segment is an angle bisector of a triangle if and only if a) it lies in the ray which bisects an angle of the triangle and b) its endpoints are the vertex of this angle and a point on the opposite side of that vertex. Any triangle has three angle bisectors. Note: An angle bisector and a median of a triangle are sometimes different. BM is a median and BD is an angle bisector of  ABC. M Let M be the midpoint of AC.

Any triangle has three altitudes. Definition of an Altitude of a Triangle A segment is an altitude of a triangle if and only if it has one endpoint at a vertex of a triangle and the other on the line that contains the side opposite that vertex so that the segment is perpendicular to this line. ACUTEOBTUSE B A C

RIGHT A B C If  ABC is a right triangle, identify its altitudes. BG, AB and BC are its altitudes. G Can a side of a triangle be its altitude?YES!

Definition of an Equidistant Point A point D is equidistant from B and C if and only if BD = DC. D B C If BD = DC, then we say that D is equidistant from B and C.

Theorem: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. RS T V U Let TU be a perpendicular bisector of RS. Then, what can you say about T, V and U? RT = TS RV = VS RU = US M

Theorem: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment. The converse of this theorem is also true: Theorem: If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

FG H Given: HF = HG Conclusion: H lies on the perpendicular bisector of FG.

Theorem: If two points and a segment lie on the same plane and each of the two points are equidistant from the endpoints of the segment, then the line joining the points is the perpendicular bisector of the segment. RS T V If T is equidistant from R and S and similarly, V is equidistant from R and S, then what can we say about TV? TV is the perpendicular bisector of RS.

Definition of a Distance Between a Line and a Point not on the Line The distance between a line and a point not on the line is the length of the perpendicular segment from the point to the line.

D Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. Let AD be a bisector of  BAC, P lie on AD, PM  AB at M, NP  AC at N. A B C M N P Then P is equidistant from AB and AC.

Theorem: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle. The converse of this theorem is not always true. Theorem: If a point is in the interior of an angle and is equidistant from the sides of the angle, then the point lies on the bisector of the angle.