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:

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

B)  \[\frac{2}{3}\sqrt{21}\]

C) \[2\sqrt{21}\]                

D) \[7\sqrt{21}\]

Correct Answer: B

Solution :

Length of median BM \[=\frac{1}{2}\sqrt{2(B{{C}^{2}}+B{{A}^{2}})-(A{{C}^{2}})}=\frac{1}{2}\sqrt{2(25+49)-36}\]

\[=\frac{1}{2}\sqrt{112}=\sqrt{\frac{112}{4}}=\sqrt{28}=2\sqrt{7}\]

Let\[h\]be height of tower, given\[\tan {{30}^{{}^\circ }}\] \[=\frac{h}{2\sqrt{h}}\Rightarrow h=2\sqrt{\frac{7}{3}}=\sqrt{\frac{28}{3}}.\]