Find the greatest leg of right triangle having a median only

Question

The median drawn to the hypotenuse is equal to $2\sqrt3$. Find the greatest leg.

Answer 1

Your point moves on a semi-circle radius $ r=2 \sqrt 3$ and diameter $d=4 \sqrt 3$ with side $ a = d \cos\,\theta$.

This $a$ is larger of two sides if $ \pi/4> \theta > 0 $ else it is the other side.

Answer 2

The longer leg limit is $4 \sqrt 3$ as the point B moves toward C