4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-4_332_454_201_810}
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\caption{Figure 1}
\end{figure}
Figure 1 shows the equilateral triangle \(L M N\) of side 2 cm ．The point \(P\) lies on \(L M\) such that \(L P = x \mathrm {~cm}\) and the point \(Q\) lies on \(L N\) such that \(L Q = y \mathrm {~cm}\) ．The points \(P\) and \(Q\) are chosen so that the area of triangle \(L P Q\) is half the area of triangle \(L M N\) ．
（a）Show that \(x y = 2\)
（b）Find the shortest possible length of \(P Q\) ，justifying your answer．
Mathematicians know that for any closed curve or polygon enclosing a fixed area，the ratio \(\frac { \text { area enclosed } } { \text { perimeter } }\) is a maximum when the closed curve is a circle．
By considering 6 copies of triangle \(L M N\) suitably arranged，
（c）find the length of the shortest line or curve that can be drawn from a point on \(L M\) to a point on \(L N\) to divide the area of triangle \(L M N\) in half．Justify your answer．
（6）