An easy way to find the distance between parallel lines, as we did in class...

Here are two parallel lines. They have common slope (as they must).

The distance is understood to be a perpendicular one, so we consider moving between the parallel lines along a perpendicular line. Any such line will do, but we may choose a convenient one.

If we can find the points at which the lines intersect, we can use the distance formula.  We know how to solve systems of equations to find the intersections. If we chose the perpendicular line wisely, then we aleady know one point without solving anything.)

In the example above, the two lines are given by    and  ,  which both have slope 2. Therefore any perpendicular line will have slope  . For convenience we choose the line , which clearly intersects the line at the origin .  To find the other point, we solve the system  

The solution is easy to find, and is the point  .  The distance between the lines is therefore the distance from this point to the other one, .  So, the distance is found with the distance formula.

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(There is a little trick in the simplification of the radical, where I factored out , using the fact that . Feel free to ignore it and use a method you are comfortable with.)

Created by Mathematica  (December 3, 2004)