Problem - 3039
Two sides of a triangle are 4 and 9; the median drawn to the third side has length 6. Find the length of the third side.
Let the length of the third side be $x$, then by the Apollonius’ Theorem (see # 3866) we have
$$4^2 + 9^2 = 2\times \Big(6^2 + \big(\frac{x}{2}\big)^2\Big)\implies x = \boxed{5\sqrt{2}}$$