Distance Between Two Points

Suppose two branches of a business, or two stops on a delivery route, sit at points $P(x_1, y_1)$ and $Q(x_2, y_2)$ on the coordinate plane. How far apart are they, measured as a straight line — “as the crow flies”? This is answered by the distance formula, and it follows directly from the Pythagorean theorem already familiar from geometry.

Draw the segment $PQ$. Through $P$ draw a horizontal line and through $Q$ draw a vertical line, and let them meet at a point $R$. Since $PR$ is horizontal, its length is simply the difference of the x-coordinates, $|x_2 - x_1|$; since $RQ$ is vertical, its length is the difference of the y-coordinates, $|y_2 - y_1|$. The angle at $R$ is a right angle, so triangle $PRQ$ is right-angled at $R$, with $PQ$ as its hypotenuse.

By the Pythagorean theorem,

so, taking the positive square root (a distance is never negative),

A useful special case follows by putting $x_1 = y_1 = 0$: the distance of a point $P(x, y)$ from the origin $O(0, 0)$ is

$OP = \sqrt{x^2 + y^2}$ …