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.

The correct option is 10 m.

Let us represent the height of the tower as $h$ meters. Let the base of the tower be at point $B$, and the top of the tower be at point $A$. Therefore, the segment $AB$ represents the tower of height $h$.

According to the problem, two points, say $C$ and $D$, lie on the same straight line extending from the base $B$ of the tower.
The distance of the first point $C$ from the base $B$ is:
$BC=5\text{meters}$
The distance of the second point $D$ from the base $B$ is:
$BD=20\text{meters}$

We are given that the angles of elevation of the top of the tower from these two points are complementary.
Recall that two angles are complementary if their sum is $90°$.
Let the angle of elevation from point $C$ be $\theta$.
Then, the angle of elevation from point $D$ must be:
$90°-\theta$

Now, let us analyze the two right-angled triangles formed with the tower:
1. In the right-angled triangle $ABC$, using the definition of the tangent function:
$\tan \left(\theta \right)=\frac{\text{Opposite side}}{\text{Adjacent side}}=\frac{AB}{BC}$
Substituting the values $AB=h$ and $BC=5$:
$\tan \left(\theta \right)=\frac{h}{5}$ ---- (Equation 1)

2. In the right-angled triangle $ABD$, using the definition of the tangent function:
$\tan (90°-\theta )=\frac{AB}{BD}$
Using the trigonometric identity $\tan (90°-\theta )=\cot \left(\theta \right)$, and substituting $AB=h$ and $BD=20$:
$\cot \left(\theta \right)=\frac{h}{20}$ ---- (Equation 2)

We know that the product of the tangent and cotangent functions of the same angle is equal to 1:
$\tan \left(\theta \right)\times \cot \left(\theta \right)=1$

Multiplying Equation 1 and Equation 2 gives:
$\left(\frac{h}{5}\right)\times \left(\frac{h}{20}\right)=1$
Simplify the equation:
$\frac{h^2}{100}=1$
$h^2=100$
Taking the square root of both sides (since height must be a positive quantity):
$h=\sqrt{100}=10\text{meters}$

Thus, the height of the tower is 10 m.