Rule

Shortest Distance From a Point to a Line

The distance from a point to a line is determined by the length of the perpendicular segment connecting the point to the line. This perpendicular segment is the shortest distance between the point and the line. For example, the distance between point P and line l is represented by segment PA.
Line l, point P not on l, and a perpendicular segment from point P to line l

Proof

The statement can be proven by using an indirect proof. Assume that the shortest distance from a point P to a line l is not the length of the perpendicular segment connecting the point to the line. Now suppose that PB is the shortest segment between point P and line l.

Line l, point P not on l, and a segment from point P to line l

By the Perpendicular Postulate, there is a segment from P that is perpendicular to l. Let that segment be PA.

Line l, point P not on l, and two segments from point P to line l

Notice that △ ABP is a right triangle. In this case, BP is the hypotenuse and, therefore, the longest side. In other words, PA is less than BP. PA < BP The assumption that BP is the shortest segment, but not perpendicular, connecting point P to line l must be false because there exists a shorter segment, PA, which is perpendicular to line l. Therefore, the shortest distance from a point to a line is the length of the perpendicular segment connecting the point to the line.

Extra

Distance From Point to Line Formula

The distance between the point P(x_0,y_0) and the line with the equation Ax+By+C=0 is calculated using the following formula.


d = |Ax_0+By_0+C|/sqrt(A^2+B^2)

Exercises