Pythagorean Theorem, Distance Formula, and Equation of a Circle

In my Algebraic and Geometric Proof of the Pythagorean Theorem post, we have learned that a right triangle with side lengths a and b and hypotenuse length c, the sum of the squares of a and b is equal to the square of c. Placing it in equation form we have c^2 = a^2 + b^2.

If we place the triangle in the coordinate plane, having A and B coordinates of (x_1,y_1) and (x_2,y_2) respectively, it is clear that the length of AC is |x_2 - x_1| and the length of BC is |x_2 - x_1|.  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,c^2 = |x_2 - x_1|^2 + |y_2-y_1|^2, 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,

c = \sqrt{|x_2 - x_1|^2 + |y_2-y_1|^2}

We say that c is the distance between A and B, and we call the formula above, the distance formula. » Read more

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