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:- What is the distance between the points (5, 6) and (– 12, 40) ?
We apply the distance formula:
- 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).