To solve for the height of an equilateral triangle with a given side length, we can use the relationship between the side length and the height of the equilateral triangle.
When dealing with an equilateral triangle, all three sides are equal, and the formula for the height [tex]\(h\)[/tex] of the triangle can be derived from the properties of the 30-60-90 right triangles that form when you draw an altitude from one vertex to the midpoint of the opposite side. The key relationship for the height [tex]\(h\)[/tex] with side length [tex]\(a\)[/tex] is:
[tex]\[ h = \frac{a \sqrt{3}}{2} \][/tex]
Here, the side length [tex]\(a\)[/tex] of our triangle is [tex]\(6\)[/tex] cm.
Plugging it into the formula, we get:
[tex]\[ h = \frac{6 \sqrt{3}}{2} \][/tex]
Simplifying this:
[tex]\[ h = 3 \sqrt{3} \][/tex]
So, the height of the triangle is [tex]\(3 \sqrt{3} \)[/tex] cm.
Among the given options, [tex]\(3 \sqrt{3} \)[/tex] cm matches with:
[tex]\[ \boxed{3 \sqrt{3} \text{ cm}} \][/tex]
