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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 .
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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 S W V T U
Practice • Pg 260 11-16 and 19-22