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. 3 square meters 9 square meters 28 square meters 34 square meters

Explanation:

Step1: Find the third - side length

Let the sides of the triangle be \(a = 7\), \(b = 6\), and the perimeter \(P=a + b + c=16\). Then \(c=P-(a + b)=16-(7 + 6)=3\).

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

The semi - perimeter \(s=\frac{a + b + c}{2}=\frac{16}{2}=8\).

Step3: Apply Heron's formula

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

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

\]

Answer:

9 square meters