Question

The angle of elevation of the top of a mountain observed from each of the three points $A,B\text{and}C$ forming a triangle $ABC$ on the plane is $\alpha$. The height of the mountain is

• A. $\left(\frac{ɑ}{1}\right)\tan ɑcotA$
• B. $\left(\frac{ɑ}{2}\right)cotɑ\tan A$
• C. $\left(\frac{ɑ}{2}\right)\tan ɑ\cos ecA$
• D. $\left(\frac{ɑ}{2}\right)\cos ecɑ\tan A$

Answer 1

The correct option is C

$\left(\frac{ɑ}{2}\right)\tan ɑ\cos ecA$

Explanation for the correct option.

Finding the height of the mountain

Given: The angle of elevation of the top of a mountain observed from each of the three points $A,B\text{and}C$ forming a triangle $ABC$ on the plane is $\alpha$

Here, the Height of the tower$=PQ$

Circumradius $=QA=QB=QC=R$

We know $\tan \theta =\frac{\text{opposite}}{\text{adjacent}},$So

In triangle $QAP,$ we get

$$\begin{gathered} tan(\angle QAP)=\frac{QP}{QA} \\ \Rightarrow \tan \alpha =\frac{QP}{R} \\ \Rightarrow QP=R\tan \alpha \end{gathered}$$

$\therefore$ Height of tower $=R\tan \alpha .$

$$\begin{gathered} R=\left(\frac{ɑ}{2sinA}\right) \\ \therefore h=\left(\frac{ɑ}{2}\right)\tan ɑ\cos ecA \end{gathered}$$

Hence the correct answer is option (C)