Question

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

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

Answer 1

Answer: (D)

$A (\hat{ j }) \cdot B (10 \hat{ i })$
$H (\hat{ h } \hat{ j }+10 \hat{ k })$
$G (10 \hat{ i }+ h \hat{ j }+10 \hat{ k })$
$\overrightarrow{ AG }=10 \hat{ i }+ hj +10 \hat{ k }$
$\overrightarrow{ BH }=-10 \hat{ i }+ h \hat{ j }+10 \hat{ k }$
$\cos \theta=\frac{\overrightarrow{ AG } \overrightarrow{ BH }}{|\overrightarrow{ AG }||\overrightarrow{ BH }|}$
$\frac{1}{5}=\frac{ h ^{2}}{ h ^{2}+200}$
$4 h^{2}=200$
$ \Rightarrow h=5 \sqrt{2}$