Question

triangle lmn has sides measuring 7 meters and 6 meters and a perimeter of 16 meters. herons formula: area = \sqrt{s(s - a)(s - b)(s - c)} what is the area of triangle lmn? round to the nearest square meter. \bigcirc 3 square meters \bigcirc 9 square meters \bigcirc 28 square meters \bigcirc 34 square meters

Explanation:

Step1: Find the third side

The perimeter of a triangle is the sum of its three sides. Let the sides be \(a = 7\), \(b = 6\), and \(c\) (the third side). The perimeter \(P=a + b + c=16\). So, \(c=16-(7 + 6)=16 - 13 = 3\) meters.

Step2: Calculate the semi - perimeter \(s\)

The semi - perimeter \(s=\frac{a + b + c}{2}\). Substituting \(a = 7\), \(b = 6\), and \(c = 3\) into the formula, we get \(s=\frac{7+6 + 3}{2}=\frac{16}{2}=8\) meters.

Step3: Apply Heron's formula

Heron's formula is \(Area=\sqrt{s(s - a)(s - b)(s - c)}\). Substitute \(s = 8\), \(a = 7\), \(b = 6\), and \(c = 3\) into the formula:
\[

$$\begin{align*} Area&=\sqrt{8(8 - 7)(8 - 6)(8 - 3)}\\ &=\sqrt{8\times1\times2\times5}\\ &=\sqrt{80}\\ &\approx9 \end{align*}$$

\]

Answer:

9 square meters