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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?


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 

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 

- Doctor Jerry, The Math Forum 
Associated Topics:
High School Coordinate Plane Geometry
High School Triangles and Other Polygons

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