Point-Line Distance (2-Dimensional) — Definition, Formula & Examples

Point-line distance is the shortest (perpendicular) distance from a given point to a line in the coordinate plane. You calculate it using the line's equation in standard form and the point's coordinates.

Given a point $(x_0, y_0)$ and a line $Ax + By + C = 0$ in $\mathbb{R}^2$ , the distance from the point to the line is defined as $d = \dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$ , which equals the length of the perpendicular segment from the point to the line.

Key Formula

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

Where:

$d$ = Perpendicular distance from the point to the line
$(x_0, y_0)$ = Coordinates of the given point
$A, B, C$ = Coefficients in the line's standard form Ax + By + C = 0

How It Works

First, rewrite the line's equation in standard form

Ax + By + C = 0

. Then substitute the point's coordinates into the numerator, take the absolute value, and divide by the square root of the sum of the squared coefficients of

x

and

y

. The result is always non-negative. If the point lies on the line, the distance is zero.

Worked Example

Problem: Find the distance from the point (1, 2) to the line 3x + 4y − 5 = 0.

Identify values: The line is already in standard form with A = 3, B = 4, C = −5. The point is (x₀, y₀) = (1, 2).

Substitute into the formula: Plug the values into the numerator and denominator.

d = \frac{|3(1) + 4(2) + (-5)|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 8 - 5|}{\sqrt{9 + 16}}

Simplify: Evaluate the absolute value and the square root.

d = \frac{|6|}{\sqrt{25}} = \frac{6}{5}

Answer: The distance from (1, 2) to the line 3x + 4y − 5 = 0 is

\dfrac{6}{5} = 1.2

units.

Why It Matters

This formula appears frequently in analytic geometry courses and standardized tests. It also underpins real applications like finding how far a GPS coordinate is from a road modeled as a line, and it generalizes to point-plane distance in 3D for multivariable calculus.

Common Mistakes

Mistake: Forgetting to rearrange the line equation so that the right side equals zero before reading off A, B, and C.

Correction: Always rewrite the equation as Ax + By + C = 0 first. For example, y = 2x + 3 becomes 2x − y + 3 = 0, so A = 2, B = −1, C = 3.