Distance Formula

In this lesson, the distance between two points whose coordinates are known will be found. A general formula for this will be developed and used.

Suppose it is desired to calculate the distance d from the point (1, 2) to the point (3, -2) shown on the grid below.

We notice that the segment connecting these points is the hypotenuse of a right triangle and use the Pythagorean Theorem. The sides can be measured by counting the grids or by subtracting the coordinates.

The vertical side of this triangle has length 4 which can be seen by subtracting the second coordinates 2 – (-2) = 4 The horizontal side has length 2 which can be seen by subtracting the first coordinates 1 – 3 = - 2 which we change to + 2 because length, or distance, is positive. Using the Pythagorean Theorem we have . Therefore . We can summarize this as follows:

The Distance Formula:

We can generalize the method used above. The distance between any two points is given by . This is known as “the distance formula.”

Let's practice:

1. What is the distance between the points (5, 6) and (– 12, 40) ?

We apply the distance formula:

2. If the distance from the point (1, 2) to the point (3, y) is , what is the value of y?

We apply the distance formula:



Squaring both sides of the final equation gives us



We solve this by factoring: which gives us two possible answers for y:
y = 0 and y = 4. Consequently there are two possible points which are located at the required distance from our given point (1, 2). They are (3, 0) and (3, 4).