To find the area of triangle LMN, given that its sides measure 7 meters, 6 meters, and the perimeter is 16 meters, we can follow these steps:
1. Identify the third side:
We know the perimeter of the triangle is 16 meters. The sum of all sides of the triangle should be equal to its perimeter. Therefore, we can find the third side by subtracting the given sides from the perimeter.
[tex]\[
\text{Third side} = 16 - 7 - 6 = 3 \text{ meters}
\][/tex]
2. Calculate the semi-perimeter:
The semi-perimeter (s) of a triangle is half of its perimeter.
[tex]\[
s = \frac{16}{2} = 8 \text{ meters}
\][/tex]
3. Apply Heron's formula to find the area:
Heron's formula for the area (A) of a triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] and semi-perimeter [tex]\( s \)[/tex] is:
[tex]\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\][/tex]
Substituting the values we have:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = 3 \)[/tex]
[tex]\[
A = \sqrt{8(8-7)(8-6)(8-3)}
\][/tex]
Simplifying inside the square root:
[tex]\[
A = \sqrt{8 \times 1 \times 2 \times 5} = \sqrt{8 \times 10} = \sqrt{80} \approx 8.944
\][/tex]
4. Round the area to the nearest square meter:
The calculated area is approximately 8.944 square meters.
Therefore, rounding to the nearest square meter, the area of triangle LMN is:
[tex]\[
9 \text{ square meters}
\][/tex]
So, the area of triangle LMN is 9 square meters.
