Question

an equilateral triangle has a semiperimeter of 6 meters. what is the area of the triangle? round to the nearest square meter. 2 square meters 7 square meters 20 square meters 78 square meters semiperimeter = 6 m herons formula: area = \sqrt{s(s - a)(s - b)(s - c)}

Explanation:

Step1: Find side length of equilateral triangle

A triangle's semiperimeter \( s=\frac{a + b + c}{2}\). For an equilateral triangle, \(a = b = c\), so semiperimeter \(s=\frac{3a}{2}\). Given \(s = 6\) meters, solve for \(a\):
\[

$$\begin{align*} 6&=\frac{3a}{2}\\ 3a&=12\\ a& = 4 \end{align*}$$

\]
So each side \(a=b=c = 4\) meters.

Step2: Apply Heron's formula

Heron's formula: \( \text{Area}=\sqrt{s(s - a)(s - b)(s - c)}\). Substitute \(s = 6\), \(a = 4\), \(b = 4\), \(c = 4\):
\[

$$\begin{align*} \text{Area}&=\sqrt{6(6 - 4)(6 - 4)(6 - 4)}\\ &=\sqrt{6\times2\times2\times2}\\ &=\sqrt{48}\\ &\approx6.928 \end{align*}$$

\]
Rounded to the nearest square meter is \(7\).

Answer:

7 square meters