The area of an equilateral triangle is 9√3 square centimetres. What is the perimeter of the triangle?

Introduction / Context:
This problem gives the area of an equilateral triangle and asks for its perimeter. The key is to use the area formula for an equilateral triangle to recover the side length and then multiply by three to obtain the perimeter. This tests knowledge of standard formulas and algebraic manipulation with square roots.

Given Data / Assumptions:

• The triangle is equilateral.
• Area A = 9√3 square centimetres.
• Let the side length be a cm.
• Perimeter P = 3a.

Concept / Approach:
The area of an equilateral triangle with side a is A = (sqrt(3) / 4) * a^2. Given the area, we can set this equal to 9√3 and solve for a. Once a is found, the perimeter is P = 3a. This requires careful handling of the square root and constants in the equation.

Step-by-Step Solution:
Step 1: Write the area formula: A = (sqrt(3) / 4) * a^2.Step 2: Set A equal to the given area: (sqrt(3) / 4) * a^2 = 9 * sqrt(3).Step 3: Divide both sides by sqrt(3): a^2 / 4 = 9.Step 4: Multiply both sides by 4: a^2 = 36.Step 5: Take the positive square root (since side length is positive): a = 6 cm.Step 6: Perimeter P = 3 * a = 3 * 6 = 18 cm.

Verification / Alternative check:
To verify, we can substitute a = 6 cm back into the area formula.Compute A = (sqrt(3) / 4) * a^2 = (sqrt(3) / 4) * 36 = 9 * sqrt(3) square centimetres, which matches the given area.Thus, the side length and perimeter are correctly determined.

Why Other Options Are Wrong:
If the perimeter were 9 cm, the side length would be 3 cm and the area would be much smaller than 9√3.If the perimeter were 9√3 cm, the side length would be 3√3 cm, leading to a different area.A perimeter of 6√3 cm similarly corresponds to a side length of 2√3 cm, giving an incorrect area.Only 18 cm corresponds to side length 6 cm, which yields the correct area.

Common Pitfalls:
Some learners forget to divide by sqrt(3) correctly when isolating a^2.Another error is to misremember the area formula and omit the factor 1 / 4, leading to an incorrect computation of a.Confusing perimeter with area and trying to directly manipulate 9√3 as if it were a length rather than an area can also cause mistakes.

Final Answer:
The perimeter of the equilateral triangle is 18 cm.