Associated Topics || Dr. Math Home || Search Dr. Math

### 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/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search