Question:
Consider a triangular plot $\mathrm{ABC}$ with sides $\mathrm{AB}=7 \mathrm{~m}, \mathrm{BC}=5 \mathrm{~m}$ and $\mathrm{CA}=6 \mathrm{~m}$. A vertical lamp-post at the mid point $\mathrm{D}$ of $\mathrm{AC}$ subtends an angle $30^{\circ}$ at $\mathrm{B}$. The height (in $\mathrm{m}$ ) of the lamp-post is:
Correct Option: , 2
Solution:
$\mathrm{BD}=\mathrm{h} \cot 30^{\circ}=\mathrm{h} \sqrt{3}$
So, $\left.7^{2}+5^{2}=2(\mathrm{~h} \sqrt{3})^{2}+3^{2}\right)$
$\Rightarrow 37=3 h^{2}+9$
$\Rightarrow 3 h^{2}=28$
$\Rightarrow \mathrm{h}=\sqrt{\frac{28}{3}}=\frac{2}{3} \sqrt{21}$