Perpendicular Distance - Proof (Two Dimensions)

Proof (Two Dimensions)

Consider the line given by and a point . For ease, consider a point given by, where (since is on the line). Then we have

.

Here is simply the angle between the line and the -axis, such that

.

Using the Pythagorean theorem we have, and

.

This can be extended to the case where is any point on the line, however, one can always choose such that one coordinate is common to to simplify the formulation.