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

**Figure 1 **Finding the distance from *A* to *C.*

To find *AB* or *BC*, only simple subtracting is necessary.

To find *AC*, though, simply subtracting is not sufficient. Triangle *ABC* is a right triangle with AC the hypotenuse. Therefore, by the *Pythagorean Theorem*,

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 *AB* = ( *x* _{2} − *x* _{1}) and *BC* = ( *y* _{2} − *y* _{1}).

Then

This is stated as a theorem.

*Theorem 101:* If the coordinates of two points are ( *x* _{1}, *y* _{1}) and ( *x* _{2}, *y* _{2}), then the distance, *d*, between the two points is given by the following formula *(Distance Formula).*

**Example 1:** Use the *Distance Formula* to find the distance between the points with coordinates (−3, 4) and (5, 2).

**Example 2:** A triangle has vertices *A*(12,5), *B*(5,3), and *C*(12, 1). Show that the triangle is isosceles.

By the *Distance Formula,*

Because *AB* = *BC,* triangle *ABC* is isosceles.