question_answer

Consider a triangular plot ABC with sides\[AB=7\,m\], \[BC=5\,m\] and\[CA=6\,m\]. A vertical lamp-post at the mid-point D of AC subtends an angle \[30{}^\circ \] at B. The height (in m) of the lamp-post is- [JEE Main Online Paper (Held On 10-Jan-2019 Morning]

A) \[2\sqrt{21}\]                            

B) \[\frac{3}{2}\sqrt{21}\]

C) \[7\sqrt{3}\]                  

D)                  \[\frac{2}{3}\sqrt{21}\]

Correct Answer: D

Solution :

Median \[{{m}_{B}}=\frac{1}{2}\sqrt{2{{a}^{2}}+2{{c}^{2}}-{{b}^{2}}}\]= \[=\,\,\frac{1}{2}\sqrt{2(49)+2(25)-36}\,=\,\sqrt{28}\] \[\tan \,30{}^\circ \,=\,\frac{h}{BD}\,=\,\frac{1}{\sqrt{3}}\] \[\Rightarrow \,\,h=\,\frac{\sqrt{28}}{\sqrt{3}}\,=\,\frac{2\sqrt{21}}{3}\,cm\]