5-4 Medians and Altitudes Course: Geometry pre-IB Quarter: 2nd Objective: Use properties of medians and altitudes of a triangle. SSS: MA.912.G.4.2, MA.912.G.4.5 Motivating the Lesson: Demonstrate how to balance a cardboard triangle on a pencil. Ask students what they think about the “balancing point” of the triangle (it’s the center). Does every triangle have this balancing point?

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).

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, divide the sum of the x-coordinates of the vertices of the triangle by 3. Do the same with the y-coordinates.

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

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.

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

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

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