Distance From a Point to a Line

Date: 09/16/98 at 00:59:43
From: Marion Keehn
Subject: Distance from a point to a line

Dear Dr. Math,
 
I'm trying to derive an equation that will give the distance from any 
point on a 2d plane to a line. I've been trying to make a line that 
crosses the original line and goes through the point with the negative 
inverse slope and the coordinates of the point. Then I'm using the 
Pythagorean formula to find the distance. I'm having trouble getting 
all the math right, so I was wondering if you might help me through it.

Thanks,
Marion Keehn

Date: 09/16/98 at 01:52:29
From: Doctor Schwa
Subject: Re: Distance from a point to a line

I also have done this problem the way you suggest, and the algebra is 
rather messy. I eventually got the answer, but when I saw how simple it 
was, I knew there had to be a better way to get there.

Probably an easier way to do this problem is to use similar triangles. 
If the original line is ax + by + c = 0, and the point we're looking at 
is (r, s), then the key triangle is the one you've identified 
(hypotenuse being a line parallel to the y-axis and down to touch the 
line, one leg being the perpendicular distance we want) and the other 
triangle is the one that shows the slope of the line.

Putting those two together, and using similar triangles, I think
you will find that the distance is |ar + bs + c| / sqrt(a^2 + b^2).

If you need help finishing off the calculation, please write back.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/