The Pythagorean theorem states that if a right triangle has legs of length $a$ and $b$, and a hypotenuse of length $c$, then $a^{2}+b^{2}=c^{2}$. Use the Pythagorean theorem and the definition of the distance between two real numbers to prove that the distance between any two points $P=\left(x_{1}, y_{1}\right)$ and $Q=\left(x_{2}, y_{2}\right)$ in the plane is given by the "distance formula" $\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$. (Hint: Draw an example of two points $P$ and $Q$ in the plane, label their coordinates, and use an appropriate right triangle.)