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:

1. 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).

Examples
 What is the distance between (–2, 7) and (4, 6)? What is your answer?
 If the distance from (x, 3) to (4, 7) is , what is the value of x? What is your answer?

M Ransom

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