To find the area of an equilateral triangle with a side length of [tex]\(5\)[/tex] cm, we can use the formula for the area of an equilateral triangle:
[tex]\[ \text{Area} = \frac{\sqrt{3}}{4} \times (\text{side length})^2 \][/tex]
In this case, the side length is [tex]\(5\)[/tex] cm. Let's apply the given values to the formula:
1. Square the side length:
[tex]\[
(\text{side length})^2 = 5^2 = 25
\][/tex]
2. Multiply by [tex]\(\frac{\sqrt{3}}{4}\)[/tex]:
[tex]\[
\text{Area} = \frac{\sqrt{3}}{4} \times 25
\][/tex]
3. Perform the multiplication:
[tex]\[
\text{Area} = \frac{25 \sqrt{3}}{4}
\][/tex]
We know from computation that the numerical value of this expression is:
[tex]\[ 10.825317547305483 \][/tex]
Thus, the closest option that matches this result is the given numerical value for the area of the equilateral triangle, which is approximately [tex]\(10.83 \ \text{cm}^2\)[/tex].
However, let's match this with the given options:
- [tex]\( \frac{5 \sqrt{3}}{4} \)[/tex] cm[tex]\(^2\)[/tex]
- [tex]\( 6.25 \)[/tex] cm[tex]\(^2\)[/tex]
- [tex]\( 6.25 \sqrt{3} \)[/tex] cm[tex]\(^2\)[/tex]
- [tex]\( 12.5 \)[/tex] cm[tex]\(^2\)[/tex]
Upon close inspection, none of the options exactly match our numerical result [tex]\(10.83 \ \text{cm}^2\)[/tex].
Therefore, based on the exact method we used and the final value calculated, it appears that there may be a mistake or an oversight in the provided answer choices. Ideally, the proper selection should have matched the computed result.
Conclusively, based on our calculation:
[tex]\[ \boxed{10.83 \ \text{cm}^2} \][/tex]
