To find the area of an equilateral triangle with a given semiperimeter of 6 meters, we'll follow these steps:
1. Determine the side length (a):
For an equilateral triangle, all three sides are equal. The semiperimeter [tex]\( s \)[/tex] is given by half the perimeter of the triangle, or:
[tex]\[
s = \frac{3a}{2}
\][/tex]
Solving for the side length [tex]\( a \)[/tex], we get:
[tex]\[
a = \frac{2s}{3}
\][/tex]
Substituting [tex]\( s = 6 \)[/tex] into the equation:
[tex]\[
a = \frac{2 \times 6}{3} = 4 \, \text{meters}
\][/tex]
2. Apply Heron's formula:
Heron's formula for the area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\][/tex]
Since this is an equilateral triangle, all sides are equal (i.e., [tex]\( a = b = c \)[/tex]). Hence:
[tex]\[
A = \sqrt{s(s-a)(s-a)(s-a)}
\][/tex]
3. Substitute the known values:
Using [tex]\( s = 6 \)[/tex] and [tex]\( a = 4 \)[/tex]:
[tex]\[
A = \sqrt{6 \times (6-4) \times (6-4) \times (6-4)}
\][/tex]
Simplify inside the square root:
[tex]\[
A = \sqrt{6 \times 2 \times 2 \times 2} = \sqrt{6 \times 8} = \sqrt{48}
\][/tex]
4. Calculate the area:
[tex]\[
A = \sqrt{48} \approx 6.928203230275509 \, \text{square meters}
\][/tex]
5. Round the area to the nearest square meter:
[tex]\[
\text{Rounded area} \approx 7 \, \text{square meters}
\][/tex]
Therefore, the area of the equilateral triangle is approximately 7 square meters. The correct choice from the given options is 7 square meters.
