Medians

Video lesson · from 48:231 worked example

One lesson video covers all of The Straight Line, so it opens at 48:23 for this topic — not from the beginning.

A line which joins a vertex to the midpoint of the opposite side.

The three medians of a triangle meet at the centroid.

A median goes to a midpoint: so find the midpoint of the opposite side first. No perpendicular gradient is involved — that is an altitude.
Use the vertex and the midpoint: the gradient is between those two points, then apply $y - b = m(x - a)$ .
Match vertex to opposite side: the median from $A$ goes to the midpoint of $BC$ .
Do not confuse the three cevians: median goes to a midpoint, altitude is perpendicular to the opposite side, perpendicular bisector does both but starts from the midpoint.

Example 1

Triangle ABC has vertices $A(4, -9)$ , $B(10, 2)$ and $C(4, -4)$ . Find the equation of the median from A.

The median from A meets the midpoint of BC.

1. Midpoint of BC:

\begin{aligned} M &= \left(\frac{10+4}{2}, \frac{2-4}{2}\right) \\ &= (7, -1) \end{aligned}

2. Gradient from A to M:

\begin{aligned} m_{AM} &= \frac{-1 - (-9)}{7 - 4} \\ &= \frac{8}{3} \end{aligned}

3. Equation (using point A):

y - (-9) = \frac{8}{3}(x - 4)

3(y + 9) = 8(x - 4)

3y + 27 = 8x - 32

3y = 8x - 59 \text{ or } 8x - 3y - 59 = 0