Two sides of a triangle have lengths $10$ and $15$.  The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides.  How long is the third side?
The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes, therefore the length of the side perpendicular to that altitude will be between $10$ and $15$. The only answer choice that meets this requirement is $\fbox{12}$.