If the altitude of an equilateral triangle is $\sqrt{3} cm$, then what is its perimeter ?

Question

Accepted Answer

To determine the perimeter of an equilateral triangle given that its altitude is $\sqrt{3} \ \text{cm}$, we need to use the properties of equilateral triangles and some basic geometry.

### Explanation Supporting Option 3:

1. **Properties of an Equilateral Triangle:**
   - In an equilateral triangle, all sides are equal, and all angles are $60^\circ$.
   - The altitude of an equilateral triangle splits it into two 30-60-90 right triangles.

2. **Using the 30-60-90 Triangle Rule:**
   - In a 30-60-90 triangle, the ratio of the sides opposite the $30^\circ$, $60^\circ$, and $90^\circ$ angles is $1 : \sqrt{3} : 2$.
   - Here, the altitude ($\sqrt{3} \ \text{cm}$) is the side opposite the $60^\circ$ angle.

3. **Finding the Side Length:**
   - Let the side length of the equilateral triangle be $s$.
   - The altitude divides the triangle into two right triangles, each with a base of $\frac{s}{2}$ and an altitude of $\sqrt{3} \ \text{cm}$.
   - According to the 30-60-90 triangle ratio, $\frac{s}{2} : \sqrt{3} = 1 : \sqrt{3}$.
   - Therefore, $\frac{s}{2} = 1 \ \text{cm}$.
   - Solving for $s$, we get $s = 2 \ \text{cm}$.

4. **Calculating the Perimeter:**
   - The perimeter of the equilateral triangle is $3s = 3 \times 2 = 6 \ \text{cm}$.

Thus, the correct perimeter of the equilateral triangle is $6 \ \text{cm}$, which corresponds to **Option 3**.

### Explanation for Why Other Options are Incorrect:

- **Option 1 (3 cm):** This option suggests that the perimeter is 3 cm, which would imply each side is 1 cm. However, given the altitude is $\sqrt{3} \ \text{cm}$, this does not satisfy the 30-60-90 triangle ratio.

- **Option 2 ($3\sqrt{3} \ \text{cm}$):** This option suggests a perimeter that does not align with the calculated side length of 2 cm. The perimeter must be a multiple of the side length, which is not the case here.

- **Option 4 ($6\sqrt{3} \ \text{cm}$):** This option implies a perimeter that is much larger than the calculated 6 cm. It suggests each side is $2\sqrt{3} \ \text{cm}$, which contradicts the given altitude and the 30-60-90 triangle ratio.

In conclusion, the correct perimeter of the equilateral triangle, given the altitude of $\sqrt{3} \ \text{cm}$, is indeed 6 cm, making **Option 3** the correct choice.