If the area of an equilateral triangle is `12sqrt(3)` sq. metre, what is the value (in metres) of its height?

Options

A. 6
B. 9
C. `7.5`
D. `4.5`

Answer

To find the height of an equilateral triangle given its area, we can use the formula for the area of an equilateral triangle and the relationship between the area and height.

### Step-by-Step Solution:

1. **Recall the formula for the area of an equilateral triangle**:
The area \( A \) of an equilateral triangle with side length \( a \) can be expressed as:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
However, we can also express the area in terms of height \( h \):
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times h \]

2. **Relate the height to the side length**:
For an equilateral triangle, the height \( h \) can be expressed in terms of the side length \( a \):
\[ h = \frac{\sqrt{3}}{2} a \]

3. **Set up the equation using the given area**:
We know the area \( A = 12\sqrt{3} \) square meters. We can use the formula for the area in terms of height:
\[ A = \frac{1}{2} \times a \times h \]
Substituting \( h \) in terms of \( a \):
\[ A = \frac{1}{2} \times a \times \left(\frac{\sqrt{3}}{2} a\right) = \frac{\sqrt{3}}{4} a^2 \]
Setting this equal to the given area:
\[ \frac{\sqrt{3}}{4} a^2 = 12\sqrt{3} \]

4. **Solve for \( a^2 \)**:
To eliminate \( \sqrt{3} \), we can divide both sides by \( \sqrt{3} \):
\[ \frac{1}{4} a^2 = 12 \]
Multiplying both sides by 4:
\[ a^2 = 48 \]

5. **Find the height \( h \)**:
Now we can find the height using \( h = \frac{\sqrt{3}}{2} a \):
\[ h = \frac{\sqrt{3}}{2} \sqrt{48} \]
Simplifying \( \sqrt{48} \):
\[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \]
Therefore,
\[ h = \frac{\sqrt{3}}{2} \times 4\sqrt{3} = \frac{4 \times 3}{2} = 6 \]

6. **Conclusion**:
The height of the equilateral triangle is \( 6 \) metres.

### Final Answer:
The height of the equilateral triangle is \( 6 \) metres.