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

### Deriving the Distance Formula from the Pythagorean Theorem

```Date: 03/23/2003 at 05:14:57
From: Hans
Subject: Formula derivation

How does one derive the distance formula from the Pythagorean theorem?

Thanks.
```

```
Date: 03/23/2003 at 09:23:14
From: Doctor Jerry
Subject: Re: Formula derivation

Hi Hans,

Suppose we have points (x1,y1) and (x2,y2) in the (x,y)-plane. To keep
things simple, let's suppose that (x2,y2) is upward and to the right
of (x1,y1).

From (x1,y1) draw a horizontal line to the right, stopping just below
(x2,y2). From (x2,y2) draw a vertical line downward, stopping at
(x2,y1), where it touches the first line. Now draw a line connecting
(x1,y1) and (x2,y2).

You will see on your paper a right triangle, with right angle at
(x2,y1).

The lengths of the sides of this right triangle are x2-x1 and y2-y1.
So, the length d of the hypotenuse is (by the Pythagorean theorem)

d^2 = (x2-x1)^2 + (y2-y1)^2.

Now all you have to do is to take the positive square root of both
sides.

- Doctor Jerry, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Triangles and Other Polygons

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