A triangle with three equal sides has its area equal to $4\sqrt{3}$ square cm. What will be the perimeter of this triangle?

Question

Accepted Answer

To determine the perimeter of the triangle, we need to use the information given about the triangle being equilateral and having an area of $4\sqrt{3}$ square cm.

1. **Understanding the Properties of an Equilateral Triangle:**
   - An equilateral triangle has all three sides equal and all internal angles equal to 60 degrees.
   - The formula for the area of an equilateral triangle with side length $a$ is given by:
     $$
     \text{Area} = \frac{\sqrt{3}}{4}a^2
     $$

2. **Using the Given Area to Find the Side Length:**
   - We know the area is $4\sqrt{3}$ square cm. Setting up the equation:
     $$
     \frac{\sqrt{3}}{4}a^2 = 4\sqrt{3}
     $$
   - To solve for $a^2$, multiply both sides by 4:
     $$
     \sqrt{3}a^2 = 16\sqrt{3}
     $$
   - Divide both sides by $\sqrt{3}$:
     $$
     a^2 = 16
     $$
   - Taking the square root of both sides gives:
     $$
     a = 4 \text{ cm}
     $$

3. **Calculating the Perimeter:**
   - The perimeter $P$ of an equilateral triangle is given by:
     $$
     P = 3a
     $$
   - Substituting $a = 4$ cm:
     $$
     P = 3 \times 4 = 12 \text{ cm}
     $$

4. **Justification for Option 2:**
   - The calculated perimeter of the triangle is 12 cm, which matches Option 2. Therefore, Option 2 is the correct answer.

5. **Explanation of Why Other Options are Incorrect:**
   - **Option 1 (10 cm):** This would imply a side length of approximately 3.33 cm, which does not satisfy the area condition.
   - **Option 3 (14 cm):** This would imply a side length of approximately 4.67 cm, which also does not satisfy the area condition.
   - **Option 4 (16 cm):** This would imply a side length of approximately 5.33 cm, which again does not satisfy the area condition.

In conclusion, based on the calculations and the properties of an equilateral triangle, Option 2 (12 cm) is indeed the correct answer for the perimeter of the triangle.