The angles of elevation of the top of a… - CUET UG 2025 General Test PYQ

CUET UG 2025 — General Test Geometry

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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.

Official previous-year question

Held on 13 May 2025 · Verified 13 Jul 2026.

Options

A
10 m
B
15 m
C
$10\sqrt{3}$ m
D
20 m

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Solution

Consider a tower with two observation points along the same straight line from its base:

Point A: 5 meters from the base
Point B: 20 meters from the base

The angles of elevation from these points are complementary (they add up to 90°).

Let the height of the tower be $h$ meters.

Let the angle of elevation from Point A be $\alpha$ .

Since the angles are complementary, the angle of elevation from Point B is $(90° - \alpha)$ .

From Point A (5m away):

$\tan(\alpha) = \frac{h}{5}$

From Point B (20m away):

$\tan(90° - \alpha) = \frac{h}{20}$

Using the complementary angle property: $\tan(90° - \alpha) = \cot(\alpha) = \frac{1}{\tan(\alpha)}$

Therefore:

$\frac{1}{\tan(\alpha)} = \frac{h}{20}$

Since $\tan(\alpha) = \frac{h}{5}$ :

$\frac{1}{\frac{h}{5}} = \frac{h}{20}$

$\frac{5}{h} = \frac{h}{20}$

$5 \times 20 = h^2$

$100 = h^2$

$h = 10$

Therefore, the height of the tower is 10 meters.

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