Distance Formula

Start by plotting $A(x_1,y_1)$ and $B(x_2,y_2)$ on the coordinate plane. Both points can be arbitrarily plotted in Quadrant I for simplicity. Note that the position of the points in the plane does not affect the proof. Assume that $x_2$ is greater than $x_1$ and that $y_2$ is greater than $y_1.$

Next, draw a right triangle. The hypotenuse of this triangle will be the segment that connects points $A$ and $B.$ The difference between the $x\text{-}$coordinates of the points is the length of one of the legs of the triangle. Furthermore, the length of the other leg is given by the difference between the $y\text{-}$coordinates. Therefore, the lengths of the legs are $x_2-x_1$ and $y_2-y_1.$ Now, consider the Pythagorean Equation. Here, $a$ and $b$ are the lengths of the legs, and $c$ the length of the hypotenuse of a right triangle. Substitute the expressions for the legs for $a$ and $b$ to find the hypotenuse's length. Then, the equation can be solved for $c.$ Note that, when solving for $c,$ only the principal root was considered. The reason is that $c$ represents the length of a side and therefore must be positive. Keeping in mind that $c$ is the distance between $A(x_1,y_1)$ and $B(x_2,y_2),$ then $c=d.$ By the Transitive Property of Equality, the Distance Formula is obtained.