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. (1) $\frac{3}{2} \sqrt{21}$

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

  3. (3) $2 \sqrt{21}$

  4. (4) $7 \sqrt{3}$

Correct Option: , 2


Let the height of the lamp-post is $h$.

By Appolonius Theorem,

$2\left(B D^{2}+\left(\frac{A C}{2}\right)^{2}\right)=B C^{2}+A B^{2}$

$\Rightarrow 2\left(m^{2}+3^{2}\right)=25+49 \Rightarrow m=2 \sqrt{7}$

$\tan 30^{\circ}=\frac{h}{B D}$

$\Rightarrow h=2 \sqrt{7} \times \frac{1}{\sqrt{3}}=\frac{2 \sqrt{21}}{3}$

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