The angles of elevation of the top of a tower from two points at a distance of 5 meters and 20 meters along the same straight line from the base of the tower, are complementary. Find the height of the tower.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 10 m
Step 1: Let the height of the tower be h
- Two points along the same straight line from the base: $x_1 = 5m, x_2 = 20 m$
- Let the angles of elevation be θ (from 5 m) and $90^\circ - \theta$ (from 20 m) because angles are complementary
Step 2: Use tangent formula
$\tan \theta = \frac{h}{5}$
$\tan(90^\circ - \theta) = \frac{h}{20} ⟹\cot \theta = \frac{h}{20} ⟹ \frac{1}{\tan \theta} = \frac{h}{20} ⟹ \tan \theta = \frac{20}{h}$
Step 3: Equate $\tan \theta$
$\frac{h}{5} = \frac{20}{h}$
$h^2 = 100$
$h = 10 \text{ m}$