Distance Formula

In Figure 1, A has coordinates (2,2), B has coordinates (5,2), and C has coordinates (5,6).

Figure 1. A right triangle.

figure

To find the length of AB or BC, only simple subtraction is necessary.

equation

To find the length of AC, however, simple subtraction is not sufficient. Triangle ABC is a right triangle with AC being the hypotenuse. Therefore, by the Pythagorean theorem,

equation

From the Pythagorean theorem, we derive the distance formula, which is nothing more than a different format for the former. If A is represented by the ordered pair ( x 1 ,y 1) and C is represented by the ordered pair ( x 2 ,y 2), then equation

Then equation

equation

d in the preceding formula stands for distance.

Example 1

Use the distance formula to find the distance between the points with coordinates (–3,4) and (5,2).

Let (–3,4) = ( x 1 ,y 1) and (5,2) = ( x 2 ,y 2). Then equation