We can find the distance between two points using an expression derived from the Pythagorean Theorem. Suppose we have two points and and we want to find the distance between them. Remember the Pythagorean Theorem for finding the length of the hypotenuse, c, of right triangle with sides, a and b, is

Solving for c we get:

For example, take the points (-2,1) and (5,4) again. We can see their location and the slope of the line connecting them.

Now if we form a triangle using these points it looks something like this:

The length of side a is equal to the difference between the x coordinates of the points (-2,1) and (5,1). a = 5 - -2 = 7. Side b is found by taking difference between the y coordinates of the points (5,1) and (5,4). b = 4 - 1 = 3.

Now we have a and b of the pythagorean theorem and all we
need to do is some arithmetic. Remember our equation:

To get the length of side c which is the distance between our original points (-2,1) and (5,4), we substitute a and b into the above equation. which gives us a value of which is about 7.6. In summary, the final equation for finding the distance between two points, and is

The final topic for lines is the relationship between two lines on the same axes. Two lines are parallel if the have the same slope, and two lines are perpendicular if .