JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Three Dimensional Geometry

Place $A$ at the origin with the two floor edges $AB$ and $AD$ along the horizontal axes and the height $h$ along the vertical, so that $E,F,G,H$ stand directly above $A,B,C,D$ respectively. With $AB=AD=10$,

and these two space diagonals cross at $P$, so the angle $GPH$ is the angle between them:

Setting this equal to $\frac{1}{5}$ gives $5h^2=200+h^2$, so $h^2=50$ and the height of the hall is $5\sqrt{2}$ m.