Distance between 2 parallel lines (geometry)

One of the topic that geometry students will encounter is proving 2 lines are parallel or perpendicular.  Somewhere in that parallel and perpendicular section, some students will be asked to determine the distance between 2 parallel lines.  By asking for the distance between 2 parallel lines, you can assume (99%) of the time, they are asking for the "shortest" distance between 2 lines.  And the only "shortest" distance between 2 lines is when you take any point on the first line and draw a perpendicular line from this point to this first line and extend it all the way to the second line.

It will be easy to figure out the distance between 2 parallel lines if the 2 lines are parallel to the x or y axis.  For example, line #1 (x = 3) and line #2 (x = 10) are parallel to the y axis.  We will NOT need any special formula to find the distance between the 2 lines.  It is just a simple 10 - 3 = 7 (and the same goes for 2 parallel lines parallel to the x axis).   But when the 2 parallel lines are something like

Line #1:  y = 3x + 2
Line #2:  y = 3x - 2

It becomes just a little harder to solve this type of problem. 

Step #1.  Find an equation of a perpendicular line to any of the parallel lines.

For this example, I will find the equation of a perpendicular line to Line #1.  Since the slope of line #1 is 3.  The slope of a perpendicular line will be the negative reciprocal of 3, which is -1/3.  Therefore the equation of a perpendicular line to line #1 will look like

Perpendicular line:  y = -1/3x + b

We will need to find "b" (which is the y intercept). In order to do that, we will need to find an ordered pair (x, y) that will satisfy this perpendicular line.  Why not use an ordered pair on Line #1 that we know for sure, such as (0, 2), which is the y intercept of Line #1.  We can substitute (0, 2) into the perpendicular line y = -1/3x + b

2 = -1/3(0) + b
2 = b

So the equation of the perpendicular line is

Line #3:  y = -1/3x + 2

Since Line #2 and Line #3 has to intersect at some point, we are basically trying to solve the 2 linear equation.

Line #2:  y = 3x - 2
Line #3:  y = -1/3x + 2

3x - 2 = -1/3x + 2
3 1/3x = 4
x = 4/(3 1/3)
x = 6/5

Substitute x = 6/5 into either Line #2 or Line #3, you will get y = 8/5.

So Line #2 and Line #3 intersects at the point (6/5, 8/5).  We can then use the distance formula to find the distance between points (0, 2) and (6/5, 8/5). 

Distance formula = sqrt[(x2 - x1)^2 + (y2 - y1)^2]

Distance = sqrt[(6/5 - 0)^2 + (8/5 - 2)^2] = sqrt[36/25 + 4/25] = sqrt(40/25) = 2*sqrt(10)/5

Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T:     201-444-8020   
E:  glenrock@mathnasium.com
www.mathnasium.com/glenrock