Perpendicular Distance from Straight Line in Plane to Point/General Form/Proof 2

Proof
Let a perpendicular be dropped from $P$ to $\LL$ at $Q$.

Let $PQ$ make an angle $\alpha$ with the $x$-axis.

Let $p$ be the length of $PQ$.

Then the coordinates of $Q$ are given by:
 * $Q = \tuple {x_0 + p \cos \alpha, y_0 + p \sin \alpha}$

$Q$ lies on $a x + b y + c$, and so:


 * $a \paren {x_0 + p \cos \alpha} + b \paren {y_0 + p \sin \alpha} + c = 0$

But from Condition for Straight Lines in Plane to be Perpendicular:

The minus sign has no immediate significance, and the result follows.