A 10 meters high tower is standing at the centre of an equilateral triangle and each side of the triangle subtends an angle of `60^(@)` at the top of the tower. Then the length of each side of the triangle is.

Options

A. 5 m
B. `5sqrt(6)`m
C. `4sqrt(6)`m
D. 4m

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understanding the Setup
We have a tower of height 10 meters standing at the center of an equilateral triangle. Each side of the triangle subtends an angle of 60 degrees at the top of the tower.

### Step 2: Define Points
Let the vertices of the equilateral triangle be A, B, and C, and let P be the top of the tower. The height of the tower (PQ) is 10 meters.

### Step 3: Establish Angles
Since the triangle is equilateral, the angles at points A, B, and C are all 60 degrees. Therefore, the angles subtended at the top of the tower (point P) by the sides of the triangle are also 60 degrees.

### Step 4: Use Right Triangle Properties
In triangle APQ (where Q is the foot of the perpendicular from P to side AB), we can use the sine function to relate the height and the distance from the center to the vertices of the triangle.

Using the sine rule:
\[ \sin(60^\circ) = \frac{PQ}{AQ} \]
Where \(PQ = 10\) meters (height of the tower) and \(AQ\) is the distance from the center of the triangle to vertex A.

### Step 5: Calculate AQ
From the sine function:
\[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \]
Thus,
\[ \frac{\sqrt{3}}{2} = \frac{10}{AQ} \implies AQ = \frac{10}{\sin(60^\circ)} = \frac{10}{\frac{\sqrt{3}}{2}} = \frac{20}{\sqrt{3}} \text{ meters} \]

### Step 6: Relate AQ to Side Length
Since AQ is the distance from the center to vertex A, we can relate this to the side length of the triangle. In an equilateral triangle, the relationship between the side length (s) and the distance from the center to a vertex (d) is given by:
\[ d = \frac{s}{\sqrt{3}} \]
Thus,
\[ AQ = \frac{s}{\sqrt{3}} \implies \frac{20}{\sqrt{3}} = \frac{s}{\sqrt{3}} \implies s = 20 \text{ meters} \]

### Step 7: Conclusion
The length of each side of the equilateral triangle is:
\[ \boxed{20 \text{ meters}} \]