Distance Formula

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 ( x₁, y₁) and C is represented by the ordered pair ( x₂, y₂), then AB = ( x₂ − x₁) and BC = ( y₂ − y₁).

Then

This is stated as a theorem.

Theorem 101: If the coordinates of two points are ( x₁, y₁) and ( x₂, y₂), 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.

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