5-4 Medians and Altitudes

1. 5-4 Medians and Altitudes

2. Medians • A median of a triangle is a segment whose endpoint are a vertex and the midpoint of the opposite side. • The point of concurrency of the medians is the centroid of the triangle (the “balancing” point).

3. Centroid Concurrency of Medians Theorem: 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. DC = ⅔DJ EC = ⅔EG FC = ⅔FH • To find the x-coordinate of the centroid, average the x-coordinates of the vertices of the triangle. Do the same to find the y-coordinate.

4. Finding a Centroid • The coordinates of ABC are A(7, 10), B(3, 6), and C(5, 2).

5.  Find the coordinates of the centroid of a triangle whose vertices are (0, 3), (6, -2), and (3, 5).

6. Altitudes • An altitude of a triangle is the perpendicular segments from a vertex of the triangle to the line containing the opposite side. Concurrency of Altitudes Theorem: The lines that contain the altitudes of a triangle are concurrent. • The lines that contain the altitudes of a triangle are concurrent at the orthocenter • Can be INSIDE, ON, or OUTSIDE the triangle.

7. Identifying Medians and Altitudes • Is PR a median, altitude, or neither? Explain. • Is QT a median, altitude, or neither? Explain.

8.  For ABC, is each segment a median, altitude, or neither. Explain. • AD • EG • CF

9. Peanut Butter Cookies Are Better If Mom Cooks After Oprah Perp. Bisectors—Circumcenter Angle Bisectors—Incenter Medians—Centroid Altitudes—Orthocenter “Trick” for Remembering Points of Concurrency *Also, the ones “inside” are always “inside” the triangle!