To find the area of triangle LMN using the given information and Heron's formula, we can follow these steps:
1. Determine the third side of the triangle:
- Given the sides of the triangle are 7 meters and 6 meters, and the perimeter of the triangle is 16 meters.
- We know that the perimeter of a triangle is the sum of all its sides.
- Let the third side be denoted as [tex]\( c \)[/tex].
[tex]\[
a + b + c = 16
\][/tex]
Plugging in the given values:
[tex]\[
7 + 6 + c = 16
\][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[
c = 16 - 7 - 6
\][/tex]
[tex]\[
c = 3
\][/tex]
So, the sides of the triangle are 7 meters, 6 meters, and 3 meters.
2. Calculate the semi-perimeter (s):
- The semi-perimeter [tex]\( s \)[/tex] is half of the perimeter of the triangle.
[tex]\[
s = \frac{a + b + c}{2}
\][/tex]
Using the values of the sides we found:
[tex]\[
s = \frac{7 + 6 + 3}{2}
\][/tex]
[tex]\[
s = \frac{16}{2}
\][/tex]
[tex]\[
s = 8
\][/tex]
3. Use Heron's formula to calculate the area:
- Heron's formula states that the area [tex]\( A \)[/tex] of a triangle can be found using:
[tex]\[
A = \sqrt{s(s - a)(s - b)(s - c)}
\][/tex]
Plugging in the values we have:
[tex]\[
A = \sqrt{8 \left(8 - 7\right) \left(8 - 6\right) \left(8 - 3\right)}
\][/tex]
Simplify inside the square root:
[tex]\[
A = \sqrt{8 \times 1 \times 2 \times 5}
\][/tex]
[tex]\[
A = \sqrt{8 \times 10}
\][/tex]
[tex]\[
A = \sqrt{80}
\][/tex]
4. Round the area to the nearest square meter:
- Calculating the square root of 80 and rounding to the nearest whole number:
[tex]\[
\sqrt{80} \approx 8.944
\][/tex]
Rounded to the nearest square meter:
[tex]\[
\approx 9
\][/tex]
Therefore, the area of triangle LMN rounded to the nearest square meter is 9 square meters. The correct option is 9 square meters.
