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

Options

A. 5m
B. `5 sqrt6 m`
C. `4sqrt6`
D. `4m`

Answer

To solve the problem, we need to find the length of each side of an equilateral triangle given that a tower of height 10 meters is standing at its center, and each side of the triangle subtends an angle of \(60^\circ\) at the top of the tower.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the equilateral triangle be denoted as \(ABC\) with \(P\) as the center where the tower is located. The height of the tower \(PQ\) is 10 meters.
- The angles subtended by each side of the triangle at the top of the tower (point \(Q\)) are \(60^\circ\).

2. **Identifying the Triangle**:
- Since \(P\) is the centroid of the equilateral triangle, the segments \(AP\), \(BP\), and \(CP\) are equal.
- The angles \(AQB\), \(BQC\), and \(CQA\) are all \(60^\circ\).

3. **Using Triangle Properties**:
- In triangle \(APQ\), we can apply the sine rule or trigonometric ratios.
- The angle \(APQ\) is \(90^\circ\) (height) and angle \(AQP\) is \(60^\circ\).

4. **Finding Lengths**:
- Let \(s\) be the length of each side of the equilateral triangle. The distance from the center \(P\) to any vertex \(A\) is \(AP\).
- The height \(PQ\) divides the triangle into two equal parts, and we can use the properties of the triangle to find \(AP\).

5. **Using the Sine Function**:
- In triangle \(APQ\):
\[ \sin(60^\circ) = \frac{PQ}{AP} \implies \frac{\sqrt{3}}{2} = \frac{10}{AP} \implies AP = \frac{10 \cdot 2}{\sqrt{3}} = \frac{20}{\sqrt{3}}. \]

6. **Finding the Side Length**:
- The centroid divides the median in the ratio \(2:1\). Therefore, the length of the median \(AD\) (where \(D\) is the midpoint of \(BC\)) is:
\[ AD = 3 \cdot AP = 3 \cdot \frac{20}{\sqrt{3}} = \frac{60}{\sqrt{3}}. \]
- The length of each side \(s\) of the equilateral triangle can be found using the formula for the median of an equilateral triangle:
\[ AD = \frac{s \sqrt{3}}{2} \implies \frac{60}{\sqrt{3}} = \frac{s \sqrt{3}}{2} \implies s = \frac{60 \cdot 2}{3} = 40. \]

### Final Answer:
The length of each side of the equilateral triangle is \(40\) meters.