Question

Consider a triangular plot ABC with sides AB = 7 m, BC = 5 m and CA = 6 m. A vertical lamp-post at the midpoint D of AC subtends an angle ${30}^{\circ }$ at B. The height (in m) of the lamp-post is:

• A. $7\sqrt{3}$
• B. $\frac{2}{3}\sqrt{21}$
• C. $\frac{3}{2}\sqrt{21}$
• D. $2\sqrt{21}$

Answer 1

The correct option is B $\frac{2}{3}\sqrt{21}$

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

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

$49+25=2(3h^2+9)$

$74=2(3h^2+9)$

$37=3h^2+9$

$37-9=3h^2$

$28=3h^2$

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

$\Rightarrow h=\sqrt{\frac{28}{3}}$

$=\frac{2\sqrt{7}}{\sqrt{3}}\times \frac{\sqrt{3}}{\sqrt{3}}$

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