question_answer

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

Answer:

Let height AB of tower = h.

In \[\Delta \,ABC\],

\[\frac{AB}{BC}=\tan \,(90-\theta )\]

\[\frac{h}{4}=\cot \,\theta \]                                                    ?(i)

In \[\Delta \,ABD\],

\[\frac{AB}{BC}=\tan \,\theta \]

\[\frac{h}{16}=\tan \,\theta \]                                                     ?(ii)

Multiply eq. (i) and (ii)

\[\frac{h}{4}\times \frac{h}{16}=\cot \,\,\theta \,\times \tan \,\theta \]

\[\frac{{{h}^{2}}}{64}=1\]

[\[\because \] \[\cot \,\theta \times \tan \,\theta =\frac{1}{\tan \,\theta }\times \tan \,\theta =1\]]

\[\Rightarrow \]   \[{{h}^{2}}=64\Rightarrow h=8\,m\]

Height of tower\[=8\text{ }m\].