4.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-4_332_454_201_810}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\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 $L M N$ .
\begin{enumerate}[label=(\alph*)]
\item Show that $x y = 2$
\item 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,
\item 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)
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2017 Q4 [13]}}