5.3 Concurrent Lines, Medians, and Altitudes Chapter 5 Relationships Within Triangles

5.3 Concurrent Lines, Medians, and Altitudes Concurrent: When three or more lines intersect in one point Point of concurrency: The point where three or more lines intersect

5.3 Concurrent Lines, Medians, and Altitudes Theorem 5-6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices Theorem 5-7 The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides

Circumcenter Circumcenter of the triangle: The point of concurrency of the perpendicular bisectors Points Q, R, and S are equidistant from C, the circumcenter The circle is circumscribed about the triangle S C R Q Perpendicular Bisectors

Incenter The incenter of the triangle is the point of concurrency of the angle bisectors Points X, Y, and Z are equidistant from I, the incenter. The circle is inscribed in the triangle T Y Angle Bisector I X V U Z

Median of a Triangle The median of a triangle is a segment that goes from the vertex to the midpoint of the opposite side.

Theorem 5-8 The medians of a triangle are concurrent at a point that is two third the distance from each vertex to the midpoint of the opposite side 8 3 6 4

Centroid The point of concurrency of the medians is the Centroid

Altitude of a Triangle Altitude: perpendicular segment from a vertex to the line containing the opposite side. * The altitude can be inside the triangle, outside the triangle, or a leg of the triangle

Orthocenter of the Triangle The lines containing the altitudes of a triangle are concurrent at the orthocenter.

Identifying Medians and Altitudes W V T U

Practice Pg 260 11-16 and 19-22