The distance between a point P and a line

*e*on the plane is the length of the line segment that is perpendicular to the line e and that joins point P to the line.### Notation

The distance between a point P and a line *e* is: d(P, *e*), which is read “distance from P to *e*.”

### Formula

In a Cartesian coordinate system, the distance *d* between a point P with coordinates \((x_{1},y_{2})\) and a line *e* with the equation \(\mathrm{A}x+\mathrm{B}y+\mathrm{C}=0\) est donnée par la formule :

\(d(\mathrm{P},\mathrm{e})=\left | \frac{\mathrm{A}x_{1}+\mathrm{B}y_{1}+\mathrm{C} } {\sqrt{\mathrm{A}^{2}+\mathrm{B}^{2}}} \right |\)

### Example

Consider the line *e* with the equation \(3x+y-1=0\) and the point P(2, 5). The distance from P to *e* can be calculated as follows:

\(d(\mathrm{P},\mathrm{e})= \frac{3\times 2+1\times5-1 } {\sqrt{3^{2}+1^{2}}}\)

or approximately 3.1623.