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The distance formula is derived from the Pythagorean Theorem.		Figure 1
The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse: u2 + v2 = w2
Proof of the Pythagorean Theorem.
When a right triangle is put on a coordinate plane, we can use the Pythagorean Theorem to calculate the length of the hypotenuse, w.
Figure 2	First we need to find the lengths of the legs u and v.
	From the points marked on the x-axis, we see that the length of u is x2 - x1.
	Similarily, using the points marked on the y-axis, we see that the length of v is y2 - y1.
	Substituting the lengths of u and v into the Pythagorean Theorem yields the following equation: (x2 - x1)2 + (y2 - y1)2 = w2
We solve for w by taking the square root of this equation. This gives us: w = Ö _________________ (x2 - x1)2 + (y2 - y1)2 = [(x2 - x1)2 + (y2 - y1)2]½ This is the Distance Formula.