Minimum distance between two parallel lines - Part III

Alright. In this post, we do the dirty work. I've set up the question in the first post, we looked at our strategy in the second post, and here in the third post, we'll find our solution. Please refer back to those pages as you need, and then come back and follow along on this page.

To start, let's find the equation of the perpendicular line between the two parallel lines. We have a point (3,8), and we want the slope that is perpendicular to the slope (5/2)... which is (-2/5). Now we can find our equation.

(y-y1) = m(x-x1)
y-8 = (-2/5)(x-3)
y=(-2/5)x + (6/5) +8
y=(-2/5)x + (46/5) This is the equation of our perpendicular line.

Now we want to find the intersection of this perpendicular line with ! our first line (equation y=(5/2)-4). To do this, we set them equal to each other, solve for the variable x, and then sub in this x value into either one of the original line equations to find y. x and y are the ordered pair of our second coordinate.

"Solve the following system:"
y=(-2/5)x + (46/5)
y=(5/2)x - 4

(-2/5)x + (46/5) = (5/2)x - 4
(-2/5)x - (5/2)x = (-4) - (46/5)
(-4-25)x/10 = (-20-46)/5
(-29)x/10 = (-66/5)
x = (-66/5)(10) / (-29)
x = 660 / (5)(29)
x = 132/29 Here is our x value! (Kinda ugly, but sometimes that's what you get!)

Now we put that value into either one of the first two equations in our system, and solve for y. Both will give the same answer... they have to! We're talking about the POINT where the two lines cross, and hence are EQUAL.

y = (5/2)x - 4
y = (5/2)(132/29) - 4
y = ! 660/58 - 4
y = (660 - 232) / 58
y = 428/58
y = 214/29 Here is our y value! (Just as ugly as x!)

So, we have our point on the first line as (132/29 , 214/29). Now all we need to do is find the distance between this point, and the given point (3,8). (Why couldn't the first pair look like this!) To do this we can use the distance formula. I've explained the distance formula in another post, so I'll just go ahead and use it here.

distance, d = sqrt[(x2-x1)^2 + (y2-y1)^2]
(Again, my apologies for not being able to show a proper square root sign.)
d = sqrt[(3-132/29)^2 + (8-214/29)^2]
d = sqrt[(-45/29)^2 + (18/29)^2]
d = sqrt[(2025+324)/841]
d = sqrt[2349/841]
d = (9/29)*sqrt(29) This is the answer!

Unfortunately, we get a nasty looking answer. But that doesn't make it any less correct. Hopefully you were able to follow along with how I approached this problem, and how we arrived at t! he solution. As always, feel free to leave any comments or questions below, and I'll do my best to address them. :)

Angle of intersection of two curves