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

Figure 1. A right triangle.

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

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,

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

Then

*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