Each side of an equilateral triangle subtends an angle of `60^(@)` at the top of a tower h m high located at the centre of the triangle. If a is the length of each side of the triangle, then

Options

A. `3a^(2)=2h^(2)`
B. `2a^(2)=2h^(2)`
C. `a^(2)=3h^(2)`
D. `3a^(2)=h^(2)`

Answer

To solve the problem, we need to establish a relationship between the height of the tower \( h \) and the length of each side of the equilateral triangle \( a \).

### Step-by-Step Solution:

1. **Draw the Diagram**:
- Start by sketching an equilateral triangle \( ABC \) with each side of length \( a \).
- Mark the center of the triangle as point \( O \).
- Draw a vertical line from point \( O \) to point \( P \) which represents the top of the tower, with height \( h \).

2. **Identify Angles**:
- Since \( ABC \) is an equilateral triangle, each angle is \( 60^\circ \).
- The angles \( \angle PAB \), \( \angle PAC \), and \( \angle PBC \) are all \( 60^\circ \) as stated in the problem.

3. **Use Properties of Equilateral Triangle**:
- The center \( O \) of the equilateral triangle divides the angles at vertices \( A \), \( B \), and \( C \) into two equal angles of \( 30^\circ \) each.
- Therefore, \( \angle OAP = 30^\circ \).

4. **Apply Trigonometric Ratios**:
- In triangle \( OAP \), we can use the secant function:
\[ \sec(30^\circ) = \frac{OA}{DA} \]
- Here, \( DA \) is half the length of side \( a \), so \( DA = \frac{a}{2} \).
- The value of \( \sec(30^\circ) \) is \( \frac{2}{\sqrt{3}} \).

5. **Calculate \( OA \)**:
- From the secant definition:
\[ \frac{2}{\sqrt{3}} = \frac{OA}{\frac{a}{2}} \]
- Rearranging gives:
\[ OA = \frac{a}{2} \cdot \frac{2}{\sqrt{3}} = \frac{a}{\sqrt{3}} \]

6. **Use Pythagorean Theorem**:
- In triangle \( OAP \):
\[ AP^2 = OP^2 + OA^2 \]
- Here, \( AP = a \), \( OP = h \), and \( OA = \frac{a}{\sqrt{3}} \).
- Substitute these values into the equation:
\[ a^2 = h^2 + \left(\frac{a}{\sqrt{3}}\right)^2 \]

7. **Simplify the Equation**:
- Expanding \( \left(\frac{a}{\sqrt{3}}\right)^2 \) gives:
\[ a^2 = h^2 + \frac{a^2}{3} \]
- Rearranging gives:
\[ a^2 - \frac{a^2}{3} = h^2 \]
- This simplifies to:
\[ \frac{2a^2}{3} = h^2 \]

8. **Final Relation**:
- Multiplying both sides by \( 3 \) gives:
\[ 2a^2 = 3h^2 \]
- Thus, we have the final relationship:
\[ h = \sqrt{\frac{2}{3}} a \]