If the altitude of an equilateral triangle is $\sqrt{6}$ cm. Then its area is :

Question

Accepted Answer

To determine the area of an equilateral triangle given its altitude, we can use the relationship between the side length, altitude, and area of an equilateral triangle.

1. **Relationship between side length and altitude:**
   For an equilateral triangle with side length $ a $, the altitude $ h $ can be expressed as:
   $$
   h = \frac{\sqrt{3}}{2}a
   $$
   Given that the altitude $ h = \sqrt{6} $ cm, we can set up the equation:
   $$
   \sqrt{6} = \frac{\sqrt{3}}{2}a
   $$
   Solving for $ a $, we multiply both sides by 2:
   $$
   2\sqrt{6} = \sqrt{3}a
   $$
   Divide both sides by $ \sqrt{3} $:
   $$
   a = \frac{2\sqrt{6}}{\sqrt{3}} = 2\sqrt{2}
   $$

2. **Calculating the area:**
   The area $ A $ of an equilateral triangle with side length $ a $ is given by:
   $$
   A = \frac{\sqrt{3}}{4}a^2
   $$
   Substituting $ a = 2\sqrt{2} $:
   $$
   A = \frac{\sqrt{3}}{4}(2\sqrt{2})^2
   $$
   $$
   A = \frac{\sqrt{3}}{4} \times 8
   $$
   $$
   A = 2\sqrt{3} \text{ cm}^2
   $$

Thus, the area of the equilateral triangle is $ 2\sqrt{3} \text{ cm}^2 $, which corresponds to Option 1.

3. **Explanation for why other options are incorrect:**

- **Option 2: $ 2\sqrt{2} \text{ cm}^2 $**
     - This option is incorrect because the calculation of the area using the formula for an equilateral triangle does not yield $ 2\sqrt{2} \text{ cm}^2 $. The correct calculation, as shown, results in $ 2\sqrt{3} \text{ cm}^2 $.

- **Option 3: $ 3\sqrt{3} \text{ cm}^2 $**
     - This option is incorrect because it overestimates the area. The correct calculation based on the given altitude and derived side length results in a smaller area of $ 2\sqrt{3} \text{ cm}^2 $.

- **Option 4: $ 3\sqrt{2} \text{ cm}^2 $**
     - This option is incorrect because it does not match the area calculated using the side length derived from the given altitude. The correct area is $ 2\sqrt{3} \text{ cm}^2 $.

In conclusion, the correct area of the equilateral triangle, given the altitude of $ \sqrt{6} $ cm, is $ 2\sqrt{3} \text{ cm}^2 $, which is Option 1.