If the area of an equilateral triangle is `12sqrt""3m^(2)`, then what is the perimeter of this triangle?

Options

A. a)`4.5`
B. b)`6.5`
C. c)`6sqrt""3`
D. d)`4sqrt""3`

Answer

To find the perimeter of an equilateral triangle given its area, we can follow these steps:

### Step 1: Write down the formula for the area of an equilateral triangle.
The area \( A \) of an equilateral triangle with side length \( a \) is given by the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

### Step 2: Set the area equal to the given value.
According to the problem, the area of the triangle is \( 12\sqrt{3} \, m^2 \). Therefore, we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 12\sqrt{3} \]

### Step 3: Simplify the equation.
To eliminate \( \sqrt{3} \) from both sides, we can divide both sides by \( \sqrt{3} \):
\[ \frac{1}{4} a^2 = 12 \]

### Step 4: Solve for \( a^2 \).
Now, multiply both sides by 4 to isolate \( a^2 \):
\[ a^2 = 12 \times 4 \]
\[ a^2 = 48 \]

### Step 5: Find the value of \( a \).
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{48} \]
We can simplify \( \sqrt{48} \):
\[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \]

### Step 6: Calculate the perimeter of the triangle.
The perimeter \( P \) of an equilateral triangle is given by:
\[ P = 3a \]
Substituting the value of \( a \):
\[ P = 3 \times 4\sqrt{3} = 12\sqrt{3} \]

### Step 7: State the final answer.
Thus, the perimeter of the equilateral triangle is:
\[ \text{Perimeter} = 12\sqrt{3} \, m \]

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