From the top of a 25 m high cliff, the angle of elevation to the top of a tower equals the angle of depression to the foot of the tower. Find the height of the tower (in metres).

Introduction / Context:
Equal acute angles from the same observation point to the top and bottom imply equal opposite legs over the same horizontal distance. Setting equal tangents simplifies the relation between tower height and cliff height.

Given Data / Assumptions:

• Cliff height = 25 m.
• Let tower height = H and horizontal distance = d.

Concept / Approach:
Angle of elevation to tower top: tan α = (H − 25) / d. Angle of depression to tower base: tan α = 25 / d. Equality gives (H − 25)/d = 25/d ⇒ H − 25 = 25.

Step-by-Step Solution:

Verification / Alternative check:
With H = 50, the two triangles from the observation point to tower top and base are symmetric in vertical offsets (25 up vs 25 down) over the same horizontal distance, giving equal angles.

Why Other Options Are Wrong:
40, 48, 52, 45 m break the equality of tangents derived from the geometric condition.

Common Pitfalls:
Using H instead of (H − 25) for elevation; confusing depression with elevation; assuming equal sides instead of equal tangents.

Final Answer:
50 m