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 Pythagoras's Theorem, $d^2 = |x_1 - x_2|^2 + |y_1 - y_2|^2$, and the result follows.