Distance Formula

Distance Formula

How to find the distance between two points: formula, proof, example, and its solution.

Formula

The distance between two points, (x1, y1) and (x2, y2), is square root((x2 - x1)^2 + (y2 - y1)^2).

The distance between two points (x1, y1) and (x2, y2) is
(x2 - x1)2 + (y2 - y1)2.

Proof

Draw the difference of the x and y. Then use the Pythagorean theorem.

(purple) = x2 - x1, (blue) = y2 - y1
Then, by the Pythagorean theorem,
d2 = (x2 - x1)2 + (y2 - y1)2.

So d = √(x2 - x1)2 + (y2 - y1)2.

Example

Find the distance between A(-3, 1), and B(5, 4).

Put (-3, 1) and (5, 4) into the formula.