Minimum distance between two parallel lines - Part I

A lot of students seem to have difficulties in determining the minimum distance between two parallel lines. This post will hopefully explain simply how this can be done, while showing that it isn't really anything more difficult than identifying the information that you need, and then using a series of simpler concepts to get to the final solution. You will hopefully find that these problems really aren't that difficult!

Let's begin. Sometimes you will be given two lines, or an equation to start with. Here, let's start even further back...

Find the equation of the line AB that crosses through the points A(2,1) and B(4,6).

Using my previous post here as a guide, you can determine the equation of this line.

Slope m = (y2-y1) / (x2-x1) = (6-1) / (4-2) = 5/2
Y-intercept b = y-mx = 1-(5/2)*2 = (-! 4)
Therefore: y = (5/2)x - 4. This is the equation of our line. Now that we have our first line, let's develop the rest of our problem.

What is the minimum distance between line AB and a parallel line CD that passes through point C(3,8)?

Before we get to the numbers, I want to remind you of some important points that you need in order to solve this. First, recall that parallel lines, by definition, are lines that have the same slope, and so will never cross or come closer together. By extension then, the minimum distance between the two lines can be represented by a line drawn perpendicular to the two lines. (Draw two lines and convince yourself of this! The shortest line you can d! raw between the two is perpendicular!)

So, armed wi! th this knowledge, continue on to my next post where I will outline our general strategy, and then we will will solve the problem.

Angle of intersection of two curves