Question

A hall has a square floor of dimension $10m\times 10m$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is ${\cos }^{-1}\frac{1}{5},$ then the height of the hall (in meters) is

• A. $2\sqrt{10}$
• B. $5\sqrt{2}$
• C. $5\sqrt{3}$
• D. $5$

Answer 1

The correct option is B $5\sqrt{2}$


Let height be $h.$
$A\equiv (0,0,0)$
$G\equiv (10,h,10)$
$B\equiv (10,0,0)$
$H\equiv (0,h,10)$

DRs of $AG\equiv (10,h,10)$
DRs of $BH\equiv (10,-h,-10)$
$\cos \theta =\left|\frac{10\times 10+h(-h)+10(-10)}{\sqrt{{10}^2+h^2+{10}^2}\times \sqrt{{10}^2+h^2+{10}^2}}\right|$
$\Rightarrow \frac{1}{5}=\frac{h^2}{200+h^2}$
$\Rightarrow h=5\sqrt{2}$