Answer: The area of the triangle is 9 m².
Step-by-step explanation: Given that the measures of two sides of triangle LMN are 7 meters and 6 meters.
We are to find the area of the triangle if its perimeter is 16 meters.
Let, a, b and c represents the lengths of three sides of a triangle.
Then, the area of the triangle will be
[tex]A=\sqrt{s(s-a)(s-b)(s-c)},~~~\textup{where }s=\dfrac{a+b+c}{2}.[/tex]
In ΔLMN,
a = 7 m and b = 6 meters.
The perimeter of ΔLMN is 16 meters. So, we have
[tex]a+b+c=16\\\\\Rightarrow 7+6+c=16\\\\\Rightarrow 13+c=16\\\\\Rightarrow c=3.[/tex]
This implies that
[tex]s=\dfrac{a+b+c}{2}=\dfrac{7+6+3}{2}=8.[/tex]
Therefore, the area of ΔLMN is
[tex]A\\\\=\sqrt{s(s-a)(s-b)(s-c)}\\\\=\sqrt{8\times(8-7)\times(8-6)\times(8-3))}\\\\=\sqrt{8\times 1\times 2\times 5}\\\\=\sqrt{80}\\\\=8.9442\sim 9~\textup{m}^2.[/tex]
Thus, the area of the triangle is 9 m².
