A triangle with three equal sides has its area equal to `9sqrt(3)` sq cm. What will be the perimeter of this triangle?

Options

A. 10cm
B. 18cm
C. 14cm
D. 16cm

Answer

To find the perimeter of an equilateral triangle with an area of \(9\sqrt{3}\) square centimeters, we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
An equilateral triangle has all three sides equal, and its area can be calculated using the formula:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
where \(a\) is the length of a side of the triangle.

### Step 2: Set up the equation using the given area
We know the area of the triangle is \(9\sqrt{3}\) square centimeters. Therefore, we can set up the equation:
\[ \frac{\sqrt{3}}{4} a^2 = 9\sqrt{3} \]

### Step 3: Simplify the equation
To eliminate \(\sqrt{3}\) from both sides, we can divide both sides by \(\sqrt{3}\):
\[ \frac{1}{4} a^2 = 9 \]

### Step 4: Solve for \(a^2\)
Next, we multiply both sides by 4 to isolate \(a^2\):
\[ a^2 = 9 \times 4 \]
\[ a^2 = 36 \]

### Step 5: Find the value of \(a\)
Now, we take the square root of both sides to find \(a\):
\[ a = \sqrt{36} = 6 \]

### Step 6: Calculate the perimeter
The perimeter \(P\) of an equilateral triangle is given by the sum of all three sides:
\[ P = a + a + a = 3a = 3 \times 6 = 18 \]

### Final Answer
The perimeter of the triangle is \(18\) cm.

---