The Distance Between Two Points

The distance between any two points in the coordinate plane can be calculated by using the Pythagorean theorem. For example:

What is the distance between the points (1, 3) and (7, −5)?

First, draw a right triangle connecting the points.

Second, find the lengths of the two legs of the triangle by calculating the rise and the run:

The y-coordinate changes from 3 to −5, a difference of 8 (the vertical leg).

The x-coordinate changes from 1 to 7, a difference of 6 (the horizontal leg).

Third, use the Pythagorean theorem to calculate the length of the diagonal, which is the distance between the points:

	6² + 8² = c² 36 + 64 = c² 100 = c² c = 10

The distance between the two points is 10 units.

Alternatively, to find the hypotenuse, you may have recognized this triangle as a variation of a 3−4−5 triangle (specifically, a 6−8−10 triangle).

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