# A hall has a square floor of dimension 10m x10m (see the figure) and vertical walls.

Question:

A hall has a square floor of dimension $10 \mathrm{~m} \times 10 \mathrm{~m}$ (see the figure) and vertical walls. If the angle GPH between the diagonals $\mathrm{AG}$ and $\mathrm{BH}$ is

$\cos ^{-1} \frac{1}{5}$, then the height of the hall (in meters) is :

1. 5

2. $2 \sqrt{10}$

3. $5 \sqrt{3}$

4. $5 \sqrt{2}$

Correct Option: , 4

Solution:

$\mathrm{A}(\hat{\mathrm{j}}) \cdot \mathrm{B}(10 \hat{\mathrm{i}})$

$\mathbf{H}(\hat{\mathrm{h}}+10 \hat{\mathrm{k}})$

$\mathbf{G}(10 \hat{\mathrm{i}}+\mathrm{h} \hat{\mathrm{j}}+10 \hat{\mathrm{k}})$

$\overrightarrow{\mathrm{AG}}=10 \hat{\mathrm{i}}+\mathrm{h} \hat{\mathrm{j}}+10 \hat{\mathrm{k}}$

$\overrightarrow{\mathrm{BH}}=-10 \hat{\mathrm{i}}+\mathrm{h} \hat{\mathrm{j}}+10 \hat{\mathrm{k}}$

$\cos \theta=\frac{\overrightarrow{\mathrm{AG}} \overrightarrow{\mathrm{BH}}}{|\overrightarrow{\mathrm{AG}}||\overrightarrow{\mathrm{BH}}|}$

$\frac{1}{5}=\frac{h^{2}}{h^{2}+200}$

$4 \mathrm{~h}^{2}=200 \Rightarrow \mathrm{h}=5 \sqrt{2}$