The angle of elevation of the top of a tall building from the points M and N at the distances of 72 m and 128 m, respectively, from the base of building and in the same straight line with it, are complementary. The height of the building (in m) is:

Answer: Option B

Solution

Answer: Option B

Solution:

$$\begin{array}{rl}& \text{In }\Delta ABM,\\ & \tan \left(90-\theta \right)=\frac{P}{B}=\frac{AB}{72}\\ & \frac{1}{\cot \theta }=\frac{72}{AB}\\ & \tan \theta =\frac{72}{AB}\text{. }\text{. }\text{. }\text{. }\left(\text{i}\right)\\ & \text{In }\Delta ABN,\\ & \tan \theta =\frac{P}{B}=\frac{AB}{128}\text{. }\text{. }\text{. }\text{. }\left(\text{ii}\right)\\ & \text{From equation }\left(\text{i}\right)\text{ and }\left(\text{ii}\right)\\ & \frac{72}{AB}=\frac{AB}{128}\\ & A{B}^2=128\times 72\\ & AB=\sqrt{16\times 4\times 2\times 36\times 2}\\ & AB=4\times 4\times 6\\ & AB=96\end{array}$$