Distance Formulas

Three distance formulas are developed:

1. Distance between two points $(x_1,y_1),(x_2,y_2)$:

   $D=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

   (the ordinary Pythagorean distance, already familiar from earlier classes).
2. Distance from a point to a line. For $P(x_1,y_1)$ and the line $AB\colon ax+by+c=0$: draw $CD$ through $P$ parallel to $AB$, drop the perpendicular from $P$ to $AB$ meeting it at $M$, and drop the perpendicular from the origin to $AB$ meeting it at $R$ and meeting $CD$ at $Q$. If $AB$'s normal form is $x\cos\alpha+y\sin\alpha=p$ (with $\cos\alpha=\pm a/\sqrt{a^2+b^2}$, $\sin\alpha=\pm b/\sqrt{a^2+b^2}$, $p=\mp c/\sqrt{a^2+b^2}$, from comparing coefficients as in §6.3.4), then $CD$'s normal form is $x\cos\alpha+y\sin\alpha=p'$ where $p'=x_1\cos\alpha+y_1\sin\alpha$ (since $CD$ passes through $P$). The required distance is $PM=QR=OR-OQ=p-p'$, which simplifies to

   $D=\left|\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}\right|.$

3. Distance between two parallel lines $a_1x+b_1y+c_1=0$ and $a_1x+b_1y+c_2=0$ (same $a_1,b_1$):

   $D=\frac{|c_2-c_1|}{\sqrt{a_1^2+b_1^2}}$

   — proved from (ii) by taking the point $(x_1,y_1)$ on one of the lines to be the origin (or any convenient point on it). Foot of the perpendicular and the image of a point. The coordinates of the foot of the perpendicular dropped from $(x_1,y_1)$ to $ax+by+c=0$ satisfy the parametric relation (a direct application of §6.3.3's parametric form, moving a signed distance along the direction perpendicular to the line)

   $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{-(ax_1+by_1+c)}{a^2+b^2},$

   and the image (mirror reflection) of $(x_1,y_1)$ in the same line — twice as far along the same perpendicular direction, since the foot is the midpoint of a point and its image — satisfies $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{-2(ax_1+by_1+c)}{a^2+b^2}.$ …