3.11. Distance Between Parallel Lines http://www.ck12.org
Example B
What is the shortest distance betweeny= 2 x+4 andy= 2 x−1?
Graph the two lines and determine the perpendicular slope, which is−^12. Find a point ony= 2 x+4, let’s say (-1,
2). From here, use the slope of the perpendicular line to find the corresponding point ony= 2 x−1. If you move
down 1 from 2 and over to the right 2 from -1, you will hity= 2 x−1 at (1, 1). Use these two points to determine
the distance between the two lines.
d=√
( 1 + 1 )^2 +( 1 − 2 )^2
=
√
22 +(− 1 )^2
=
√
4 + 1
=
√
5 ≈ 2. 24 unitsThe lines are about 2.24 units apart.
Notice that you could have used any two points, as long as they are on the same perpendicular line. For example,
you could have also used (-3, -2) and (-1, -3) and you still would have gotten the same answer.
d=√
(− 1 + 3 )^2 +(− 3 + 2 )^2
=
√
22 +(− 1 )^2
=
√
4 + 1
=
√
5 ≈ 2. 24 unitsExample C
Find the distance between the two parallel lines below.
First you need to find the slope of the two lines. Because they are parallel, they are the same slope, so if you find the
slope of one, you have the slope of both.
Start at they−intercept of the top line, 7. From there, you would go down 1 and over 3 to reach the line again.
Therefore the slope is−^13 and the perpendicular slope would be 3.
Next, find two points on the lines. Let’s use they−intercept of the bottom line, (0, -3). Then, rise 3 and go over 1
until your reach the second line. Doing this three times, you would hit the top line at (3, 6). Use these two points in
the distance formula to find how far apart the lines are.
d=√
( 0 − 3 )^2 +(− 3 − 6 )^2
=
√
(− 3 )^2 +(− 9 )^2
=
√
9 + 81
=
√
90 ≈ 9. 49 unitsWatch this video for help with the Examples above.