Geometry / Unit 6: Relationships in Triangles
6.4 Medians and Altitudes
Notes
- A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid, the balance point.
- Centroid Theorem: the centroid is $\displaystyle \frac{2}{3}$ of the way from each vertex to the opposite midpoint. So the vertex-to-centroid piece is twice the centroid-to-midpoint piece.
- An altitude is a perpendicular segment from a vertex to the line containing the opposite side. The altitudes meet at the orthocenter.
- Centroid on the coordinate plane: average the vertices, $\displaystyle \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$.
- Example: median $\overline{AD}$ has length 18, so the centroid $G$ gives $AG = 12$ and $GD = 6$.
Practice
$G$ is the centroid of $\triangle ABC$ and $\overline{BE}$ is a median. If $BE = 27$, find $BG$.
$\displaystyle BG = \frac{2}{3}(27) = 18$.
$G$ is the centroid of $\triangle PQR$ and $\overline{PM}$ is a median. If $GM = 5$, find $PM$.
$PG = 2 \cdot GM = 10$, so $PM = 10 + 5 = 15$.
Find the centroid of the triangle with vertices $(0, 0)$, $(6, 0)$ and $(3, 9)$.
$\displaystyle \left(\frac{0 + 6 + 3}{3}, \frac{0 + 0 + 9}{3}\right) = (3, 3)$.
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