Quadrants · Hard problem

A room shaped like the letter L has corners, in order, at $(0, 0)$, $(10, 0)$, $(10, 4)$, $(6, 4)$, $(6, H)$ and $(0, H)$, with every angle a right angle (so $H>4$). If the room's perimeter is $46$ units, what is the value of $H$?