- Find the third side length using the perimeter: \(c = P - a - b = 16 - 7 - 6 = 3\).
- Determine the semi-perimeter: \(s = P/2 = 16/2 = 8\).
- Apply Heron's formula: \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{8(8-7)(8-6)(8-3)} = \sqrt{80} \approx 8.944\).
- Round the result to the nearest square meter, yielding the final answer of \(\boxed{9}\) square meters (option a).
Explanation
- Find the third side of the triangle.
The triangle LMN has two side lengths given as \(a = 7\) meters and \(b = 6\) meters, and a known perimeter \(P = 16\) meters. The perimeter is the sum of all three side lengths of a triangle, so we can express it as \(P = a + b + c\). Substituting the given values:
\[16 = 7 + 6 + c\]
\[16 = 13 + c\]
Solving for the third side length \(c\), we get \(c = 16 - 13 = 3\) meters.
- Calculate the semi-perimeter of the triangle.
Heron's formula requires the semi-perimeter, which is half of the triangle's perimeter.
\[s = \frac{P}{2} = \frac{16}{2} = 8 \text{ meters.}\]
This value \(s\) will be used in the area calculation formula.
- Calculate the area using Heron's formula and round the result.
Heron's formula for the area of a triangle is \(\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}\). Substituting the values \(s = 8\), \(a = 7\), \(b = 6\), and \(c = 3\):
\[\text{Area} = \sqrt{8 \times (8-7) \times (8-6) \times (8-3)}\]
\[\text{Area} = \sqrt{8 \times 1 \times 2 \times 5}\]
\[\text{Area} = \sqrt{80}\]
Evaluating the square root, we find the area is approximately \(8.944\) square meters. Rounding to the nearest square meter, we get \(9\) square meters. This corresponds to option (a).
Examples
In construction, calculating the area of a non-rectangular plot of land is essential for determining how much material—like sod, gravel, or paving stones—is needed for landscaping. If you have a triangular garden plot with side lengths measuring 7m, 6m, and 3m, you can use Heron's formula to find its area. Knowing that the area is approximately 9 square meters allows you to purchase exactly the right amount of supplies, preventing both waste and shortages.