Perpendicular Distance from Straight Line in Plane to Point

Theorem

General Form

Let $\LL$ be a straight line embedded in a cartesian plane, given by the equation:

$a x + b y + c = 0$

Let $P$ be a point in the cartesian plane whose coordinates are given by:

$P = \tuple {x_0, y_0}$

Then the perpendicular distance $d$ from $P$ to $\LL$ is given by:

$d = \dfrac {\size {a x_0 + b y_0 + c} } {\sqrt {a^2 + b^2} }$

Normal Form

Let $\LL$ be a straight line in the Cartesian plane.

Let $\LL$ be expressed in normal form:

$x \cos \alpha + y \sin \alpha = p$

Let $P$ be a point in the cartesian plane whose coordinates are given by:

$P = \tuple {x_0, y_0}$

Then the perpendicular distance $d$ from $P$ to $\LL$ is given by:

$\pm d = x_0 \cos \alpha = y_0 \sin \alpha - p$

where $\pm$ depends on whether $P$ is on the same side of $\LL$ as the origin $O$.