Altitudes

Video lesson · from 39:021 worked example

One lesson video covers all of The Straight Line, so it opens at 39:02 for this topic — not from the beginning.

The altitude of a triangle is a line drawn from one vertex which meets the opposite side at right angles.

The three altitudes of a triangle meet at the orthocentre.

An altitude is perpendicular to the opposite side: so you need the gradient of the side, then its negative reciprocal.
It passes through the opposite vertex: use the vertex the altitude comes from, not a point on the side it meets.
Identify the right side: the altitude from $A$ is perpendicular to $BC$ . Mixing up which vertex pairs with which side wastes the whole question.
A sketch prevents most errors: even a rough one makes it obvious which side is opposite which vertex.

Example 1

Triangle ABC has vertices $A(3, -5)$ , $B(4, 3)$ and $C(-7, 2)$ . Find the equation of the altitude from A.

The altitude from A meets BC at right angles.

1. Gradient of BC:

\begin{aligned} m_{BC} &= \frac{2 - 3}{-7 - 4} \\ &= \frac{-1}{-11} \\ &= \frac{1}{11} \end{aligned}

2. Perpendicular Gradient:

m_{\perp} = -11

3. Equation (using point A):

y - (-5) = -11(x - 3)

y + 5 = -11x + 33

y = -11x + 28 \text{ or } 11x + y - 28 = 0