Triangles and polygons
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The sides of an equilateral triangle are \(8-2x,\,x+2\) and \(3x-2\). Find the value of \(x\) and confirm that the triangle is equilateral.
$$
\begin{aligned}
&x\ \text{and confirm that the triangle is equilateral.}\\ &\text{Let}\ A B=8-2 x,\ B C=x+2\ \text{and}\ A C=3 x-2\\
&\begin{aligned} A B=B C \quad 8-2 x &=x+2 \\ 8 &=3 x+2 \\ 3 x &=6 \\ x &=2\\
\therefore B=B C &=4 \\
A C=3 \pi-2 \\
&=6-2 \\
&=4 \\
\therefore\ \triangle\ A B C\ &\text{is equilateral}\\
\text{and}\ x&=2
\end{aligned}
\end{aligned}
$$
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