The area of an equilateral triangle is `sqrt(243)/4 sq cm`. Find the length of its side.

Options

A. 3 cm
B. 3-J3 cm
C. 9 cm
D. V3 cm

Answer

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

### Step-by-step Solution:

1. **Understand 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:
\[ A = \frac{\sqrt{3}}{4} a^2 \]

2. **Set Up the Equation**:
We are given that the area \( A \) is \( \frac{\sqrt{243}}{4} \) sq cm. We can set up the equation:
\[ \frac{\sqrt{243}}{4} = \frac{\sqrt{3}}{4} a^2 \]

3. **Eliminate the Denominator**:
To eliminate the fraction, multiply both sides by 4:
\[ \sqrt{243} = \sqrt{3} a^2 \]

4. **Square Both Sides**:
To remove the square roots, square both sides of the equation:
\[ (\sqrt{243})^2 = (\sqrt{3} a^2)^2 \]
This simplifies to:
\[ 243 = 3 a^4 \]

5. **Isolate \( a^4 \)**:
Divide both sides by 3:
\[ a^4 = \frac{243}{3} = 81 \]

6. **Solve for \( a \)**:
Take the fourth root of both sides:
\[ a = \sqrt[4]{81} \]
Since \( 81 = 3^4 \), we find:
\[ a = 3 \text{ cm} \]

### Final Answer:
The length of the side of the equilateral triangle is \( 3 \) cm.

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