The height of an equilateral triangle having each side 12 cm is :

Question

Accepted Answer

To determine the height of an equilateral triangle with each side measuring 12 cm, we can use the formula for the height of an equilateral triangle:

$$ \text{Height} = \frac{\sqrt{3}}{2} \times \text{side length} $$

Given that the side length is 12 cm, we can substitute this value into the formula:

$$ \text{Height} = \frac{\sqrt{3}}{2} \times 12 $$

$$ \text{Height} = 6\sqrt{3} \ \text{cm} $$

This calculation shows that the height of the equilateral triangle is $6\sqrt{3} \ \text{cm}$, which corresponds to Option 3.

**Explanation for why Option 3 is correct:**

1. **Formula Application:** The formula for the height of an equilateral triangle is derived from the Pythagorean theorem applied to the 30-60-90 right triangle formed by drawing an altitude from one vertex to the midpoint of the opposite side. This formula is $\frac{\sqrt{3}}{2} \times \text{side length}$, which correctly gives us $6\sqrt{3} \ \text{cm}$ when the side length is 12 cm.

2. **Mathematical Consistency:** The calculation is consistent with the properties of equilateral triangles, where the altitude divides the triangle into two 30-60-90 right triangles. The altitude (height) is always $\frac{\sqrt{3}}{2}$ times the side length.

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

1. **Option 1: $6\sqrt{6} \ \text{cm}$:** This option incorrectly applies the square root factor. The correct factor for the height of an equilateral triangle is $\sqrt{3}$, not $\sqrt{6}$.

2. **Option 2: $3\sqrt{6} \ \text{cm}$:** This option also incorrectly uses $\sqrt{6}$ instead of $\sqrt{3}$, and the coefficient is incorrect for the given side length.

3. **Option 4: $6\sqrt{2} \ \text{cm}$:** This option uses $\sqrt{2}$, which is not applicable in the context of calculating the height of an equilateral triangle. The correct factor is $\sqrt{3}$.

In conclusion, Option 3 is the correct answer because it accurately applies the formula for the height of an equilateral triangle, resulting in $6\sqrt{3} \ \text{cm}$.