Discussion
From the top of a cliff 25 m high the angle of elevation of a tower is found to be equal to the angle of depression of the foot of the tower. Find the height of the tower.
Answer: B
Let the tower be \( H \) m and \( d \) m away, and call the common angle \( \theta \).
The depression of the tower's foot uses the cliff's own height:
\[ \tan\theta = \frac{25}{d} \]
The elevation of the tower's top uses the part above the cliff:
\[ \tan\theta = \frac{H - 25}{d} \]
The two are equal, so the numerators must match:
\[ H - 25 = 25 \quad\Longrightarrow\quad H = 50 \text{ m} \]
Whenever the elevation of a top equals the depression of a foot, the object is exactly twice the observer's height - the distance never enters the answer.
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