The distance from a point to a line

The distance from a point to a line is the shortest gap, measured perpendicular to the line¹. You may join the point to any point of the line you like, but the perpendicular segment is the shortest: any slanted segment is the hypotenuse of a right triangle having the perpendicular as a leg, and a hypotenuse is longer than either leg. Here we find the distance from the origin $O(0, 0)$ to the line $3x + 4y - 25 = 0$ .

The formula

The distance from a point $(x_0, y_0)$ to a line $ax + by + c = 0$ is the following.

d = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}

Part	Value here
Numerator $\|3 \cdot 0 + 4 \cdot 0 - 25\|$	$25$
Denominator $\sqrt{3^2 + 4^2}$	$5$
Distance $d$	$5$

Reading the formula

The numerator is the absolute value of what you get by dropping the coordinates of the point into the left-hand side of the line's equation; if the point lies on the line that value is $0$ , and so is the distance. The denominator $\sqrt{a^2 + b^2}$ is the length of the vector $(a, b)$ , which points perpendicular to the line, and dividing by it converts the value of the expression into an actual length.

The foot of the perpendicular

The perpendicular dropped from $O$ to the line is $y = \dfrac{4}{3}x$ . Substituting into $3x + 4y - 25 = 0$ gives $\dfrac{25}{3}x = 25$ , so the foot is $(3, 4)$ . The length from the origin to $(3, 4)$ is $\sqrt{3^2 + 4^2} = 5$ , matching the distance from the formula.

Circles and tangency

This formula is exactly what lies behind comparing the distance from a center to a line with the radius.

Distance versus radius	The line and the circle
$d < r$	cut at two points
$d = r$	tangent
$d > r$	do not meet

The circle $x^2 + y^2 = 25$ has center $(0, 0)$ and radius $5$ , and the distance is $5$ , so this line is tangent to it.

The sign tells you the side

Without the absolute value, the sign of $a x_0 + b y_0 + c$ tells you which side of the line the point is on. Separating two sides by that sign is the idea behind the machine-learning methods that split data with a line, or a hyperplane².

The large dots on the graph are the point $O$ and the foot of the perpendicular; the length of the segment joining them is the distance we wanted.

Distance from a point to a line, Wikipedia
Support vector machine, Wikipedia