Consider a triangular plot


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:


  1. $7 \sqrt{3}$

  2. $\frac{2}{3} \sqrt{21}$

  3. $\frac{3}{2} \sqrt{21}$

  4. $2 \sqrt{21}$

Correct Option: , 2


$\mathrm{BD}=\mathrm{h} \cot 30^{\circ}=\mathrm{h} \sqrt{3}$

So, $\left.7^{2}+5^{2}=2(\mathrm{~h} \sqrt{3})^{2}+3^{2}\right)$

$\Rightarrow 37=3 h^{2}+9$

$\Rightarrow 3 h^{2}=28$

$\Rightarrow \mathrm{h}=\sqrt{\frac{28}{3}}=\frac{2}{3} \sqrt{21}$


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