The angle of depression of vertices of a regular hexagon lying in a plane from the top of a 75 m high tower standing at the centre of the hexagon is `60^(@)`. What is the length of each side of the hexagon ?

Options

A. `50sqrt(3)` m
B. 75 m
C. `25sqrt(3)` m
D. 25 m

Answer

To solve the problem step by step, we will use the information given about the regular hexagon and the height of the tower.

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a regular hexagon with vertices A, B, C, D, E, and F.
- A tower of height 75 m is located at the center of the hexagon (point O).
- The angle of depression from the top of the tower to each vertex of the hexagon is 60°.

2. **Identify the Triangle**:
- When looking from the top of the tower (point A) to a vertex of the hexagon (point B), we can form a right triangle OAB.
- In this triangle, AO is the height of the tower (75 m), and OB is the horizontal distance from the center of the hexagon to vertex B.

3. **Use the Angle of Depression**:
- The angle of depression from A to B is 60°. Therefore, the angle ∠OAB is also 60° (alternate interior angles).
- We can use the tangent function to relate the height and the distance:
\[ \tan(60°) = \frac{AO}{OB} \]
- Substituting the known values:
\[ \tan(60°) = \sqrt{3} \quad \text{and} \quad AO = 75 \text{ m} \]
\[ \sqrt{3} = \frac{75}{OB} \]

4. **Solve for OB**:
- Rearranging the equation gives us:
\[ OB = \frac{75}{\sqrt{3}} = 25\sqrt{3} \text{ m} \]

5. **Determine the Length of the Side of the Hexagon**:
- In a regular hexagon, all sides are equal, and the distance from the center to any vertex (OB) is equal to the radius of the circumcircle of the hexagon.
- The length of each side (s) of the hexagon can be calculated using the formula:
\[ s = OB \cdot \frac{\sqrt{3}}{2} \]
- Substituting the value of OB:
\[ s = 25\sqrt{3} \cdot \frac{\sqrt{3}}{2} = \frac{25 \cdot 3}{2} = \frac{75}{2} = 37.5 \text{ m} \]

6. **Final Answer**:
- The length of each side of the hexagon is \( 25\sqrt{3} \) m.