If the length of each median of an equilateral triangle is `6sqrt(3)` cm, then the perimeter of the triangle is

Options

A. 24 cm
B. 32 cm
C. 36 cm
D. 42 cm

Answer

To solve the problem, we need to determine the perimeter of an equilateral triangle given that the length of each median is \(6\sqrt{3}\) cm.

### Step-by-Step Solution:

1. **Understand the properties of the equilateral triangle:**
In an equilateral triangle, all sides are equal, and the median, altitude, and angle bisector from any vertex to the opposite side are all the same line segment.

2. **Use the formula for the length of the median:**
The length of the median \(m\) from a vertex to the midpoint of the opposite side in an equilateral triangle can be expressed as:
\[ m = \frac{\sqrt{3}}{2} \times a \]
where \(a\) is the length of a side of the triangle.

3. **Set the median equal to the given length:**
We know that the median is \(6\sqrt{3}\) cm, so we can set up the equation:
\[ \frac{\sqrt{3}}{2} \times a = 6\sqrt{3} \]

4. **Solve for \(a\):**
To isolate \(a\), we can multiply both sides of the equation by \(\frac{2}{\sqrt{3}}\):
\[ a = 6\sqrt{3} \times \frac{2}{\sqrt{3}} \]
Simplifying this gives:
\[ a = 6 \times 2 = 12 \text{ cm} \]

5. **Calculate the perimeter of the triangle:**
The perimeter \(P\) of an equilateral triangle is given by:
\[ P = 3 \times a \]
Substituting the value of \(a\):
\[ P = 3 \times 12 = 36 \text{ cm} \]

### Final Answer:
The perimeter of the triangle is \(36\) cm.