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.

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°).

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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 $(90° - \alpha)$.

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From Point A (5m away):

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

From Point B (20m away):

$\tan(90° - \alpha) = \frac{h}{20}$

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

Therefore:

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

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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.