Quadrants · Hard problem

A room shaped like an L has corners, in order, at $(0, 0)$, $(9, 0)$, $(9, 4)$, $(5, 4)$, $(5, h)$ and $(0, h)$, with every angle a right angle, where $h>4$. The area of the room is $51$ square units. What is the value of $h$?