5.
\begin{figure}[h]
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\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{c85316fe-5c59-4cb3-8cb8-d95a4e97af70-3_549_620_276_689}
\end{figure}
The shape of a badge is a sector \(A B C\) of a circle with centre \(A\) and radius \(A B\), as shown in Fig 1. The triangle \(A B C\) is equilateral and has a perpendicular height 3 cm .
- Find, in surd form, the length \(A B\).
- Find, in terms of \(\pi\), the area of the badge.
- Prove that the perimeter of the badge is \(\frac { 2 \sqrt { 3 } } { 3 } ( \pi + 6 ) \mathrm { cm }\).