Distance of a Point from a Line

5.4.2 The Distance of a Point from a Line

Theorem. The perpendicular distance of a point $P(x_1,y_1)$ from the line $ax+by+c=0$ is

Proof. This generalises the origin case. Let the line meet the axes at $B\left(-\dfrac ca,0\right)$, $C\left(0,-\dfrac cb\right)$, and let $PM=p$ be the perpendicular from $P$ to the line. Computing the area of $\triangle PBC$ two ways — once as $\tfrac12\cdot BC\cdot p$ (using $BC$ from the earlier computation), and once using the coordinate/determinant formula for the area of a triangle with vertices $P(x_1,y_1)$, $B$, $C$ — and equating the two expressions leads, after simplification, to $p=\dfrac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$. (Setting $x_1=y_1=0$ recovers exactly the origin-distance formula of 5.4.1, as it should.) $\blacksquare$ …