5.3 Concurrent Lines, Medians, and Altitudes

1. 5.3 Concurrent Lines, Medians, and Altitudes Chapter 5 Relationships Within Triangles

2. 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

3. 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

4. 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

5. 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

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

7. 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

8. Centroid • The point of concurrency of the medians is the Centroid

9. 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

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

11. Identifying Medians and Altitudes S W V T U

12. Practice • Pg 260 11-16 and 19-22