In my Algebraic and Geometric Proof of the Pythagorean Theorem post, we have learned that a right triangle with side lengths and and hypotenuse length , the sum of the squares of and is equal to the square of . Placing it in equation form we have .
If we place the triangle in the coordinate plane, having and coordinates of and respectively, it is clear that the length of is and the length of is . We are finding the length, which means that we want a positive value; the absolute value signs guarantee that the result of the operation is always positive. But in the final equation,, the absolute value sign is not needed since we squared all the terms, and squared numbers are always positive. Getting the square root of both sides we have,
We say that is the distance between and , and we call the formula above, the distance formula. » Read more