Distance Formula

Theorem
The distance $$d$$ between two points $$A=(x_1, y_1)$$ and $$B=(x_2, y_2)$$ in Cartesian coordinates is $$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$.

Proof
The distance in the horizontal direction between $$A$$ and $$B$$ is given by $$|x_1 - x_2|$$.

The distance in the vertical direction between $$A$$ and $$B$$ is given by $$|y_1 - y_2|$$.

Clearly the angle between a horizontal and a vertical line is a right angle.

So when we place a point $$C=(x_1, y_2)$$, $$\triangle ABC$$ is a right triangle.

Thus, by the Pythagorean Theorem, $$d^2 = |x_1 - x_2|^2 + |y_1 - y_2|^2$$, and the result follows.