Deriving the Distance Formula from the Pythagorean TheoremDate: 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/ |
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