A girls' camp is located $300$ rods from a straight road. On this road, a boys' camp is located $500$ rods from the girls' camp. 
It is desired to build a canteen on the road which shall be exactly the same distance from each camp. 
The distance of the canteen from each of the camps is: 
$\textbf{(A)}\ 400\text{ rods} \qquad \textbf{(B)}\ 250\text{ rods} \qquad \textbf{(C)}\ 87.5\text{ rods} \qquad \textbf{(D)}\ 200\text{ rods}\\ \textbf{(E)}\ \text{none of these}$
Let $r$ be the straight road, $G$ be the girls' camp, $B$ be the boys' camp, and $C$ be the water canteen. $\overline{AG}$ is the perpendicular from $G$ to $r$. Suppose each rod is one unit long. $AG=300$ and $BG=500$. Since $\angle GAB$ is a right angle, $\triangle GAB$ is a $3-4-5$ right triangle, so $AB=400$.
Let $x$ be the distance from the canteen to the girls' and boys' camps. We have $BC=CG=x$ and $AC=400-x$. Using the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem) on $\triangle GAC$, we have
\[300^2+(400-x)^2=x^2\] 
Simplifying the left side gives
\[90000+160000-800x+x^2=x^2\]
Subtracting $x^2$ from both sides and rearranging gives
\[800x=250000\]
Therefore, $x=\frac{250000}{800}=312.5$. The answer is $\boxed{\textbf{(E)}\ \text{none of these}}$.