Medians, Altitudes and Concurrent Lines Section 5-3

A B C Given  ABC, identify the opposite side of A of B 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. Properties of Medians

The median starts at a vertex and ends at the midpoint of the opposite side. Centroid Properties of Medians

Centroid of a Triangle: The point of concurrency of the medians of a triangle.

The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. This point of intersection is called a centroid. This point of intersection is called a centroid. D G F C J H E DC = 2/3(DJ) EC = 2/3(EG) FC = 2/3(FH) Theorem about Medians

The centroid is 2/3’s of the distance from the vertex to the side. from the vertex to the side. 2x x X16 Properties of Medians

In the figure below, DE = 6 and AD = 16. Find DB and AF. F

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. Properties of Angle Bisectors

Angle bisectors start at a vertex and bisect the angle. Incenter Properties of Angle Bisectors

Any point on an angle bisector is equidistance from the sides of the angle Properties of Angle Bisectors

This makes the Incenter an equidistance from all 3 sides Properties of Angle Bisectors

Any triangle has three (3) 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 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 Properties of Altitudes

Start at a vertex and form a 90° angle with the line containing the opposite side. Orthocenter Properties of Altitudes

The orthocenter can be located in the triangle, on the triangle or outside the triangle. Right Legs are altitudes Obtuse Properties of Altitudes

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! Properties of Altitudes

Median goes from vertex to midpoint of segment opposite. Altitude is a perpendicular segment from vertex to segment opposite. Compare Medians & Altitudes

Altitude.. Vertex.. 90°.. Orthocenter Vertex.. 90°.. Orthocenter Angle Bisector.. Angle into 2 equal angles.. Incenter Angle into 2 equal angles.. Incenter Perpendicular Bisector… 90°.. bisects side.. Circumcenter 90°.. bisects side.. Circumcenter Median.. Vertex.. Midpoint of side.. Centroid Vertex.. Midpoint of side.. Centroid

Give the best name for AB ABABABABAB || | | || Median Altitude None A Angle Perp Bisector Bisector

Concurrency Concurrent Lines: Three or more lines that meet at one point. Point of Concurrency: The point at which concurrent lines meet. l m n P k

Properties of Bisectors Theorem 5-6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Circumcenter of the Triangle: The point of concurrency of the perpendicular bisectors of a triangle.

Properties of Bisectors Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides. Incenter of the Triangle: The point of concurrency of the angle bisectors of a triangle.

Sum It Up Figure concurrent at.. which is… bisector circumcenter incenter centroid orthocenter median bisector altitude equidistant from vertices equidistant from sides 2/3 distance from vertices to midpoint