Distance of a point from a line

In analytic geometry the distance of a point from a line is computed with a direct formula from the coefficients of the equation of the line.

Property — Point-line distance

The distance of the point $P(x_0;\,y_0)$ from the line $ax+by+c=0$ is: $d(P,r) = \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}.$

Example — Distance of a point from the line

Given $A(3;1)$ and the line $s:\; 2x-3y+1=0$, compute $d(A,s)$: $d = \frac{|2\cdot 3-3\cdot 1+1|}{\sqrt{2^2+(-3)^2}} = \frac{|6-3+1|}{\sqrt{13}} = \boxed{\frac{4}{\sqrt{13}} \approx 1{,}1}$

Links

Topics: Euclidean circle
Concepts: Point-line distance · Line
Skills: Calculating · Analytic geometry · Using formulae