5. Let $n$ be a given positive integer. If a set $K$ of integer points in the plane satisfies the following condition, it is called "connected": for any pair of points $R, S \in K$, there exists a positive integer $l$ and a sequence of points in $K$
$$
R=T_{0}, T_{1}, \cdots, T_{l}=S,
$$

where the distance between each $T_{i}$ and $T_{i+1}$ is 1.
For such a set $K$, define
$$
\Delta(K)=\{\overrightarrow{R S} \mid R, S \in K\} \text {. }
$$

For all connected sets $K$ of $2 n+1$ integer points in the plane, find the maximum possible value of $|\Delta(K)|$.
5. The maximum possible value of $\mid \Delta(K)$ is $2 n^{2}+4 n+1$.

On one hand, take the set
$$
\begin{aligned}
K= & \{(0,0)\} \cup\{(0, i) \mid i=1,2, \cdots, n\} \cup \\
& \{(i, 0) \mid i=1,2, \cdots, n\},
\end{aligned}
$$

which is connected, and $|K|=2 n+1$. And
$$
\begin{aligned}
|\Delta(K)|= & \mid(0,0)\} \cup\{(0, \pm i) \mid i=1,2, \cdots, n\} \\
& \cup\{(\pm i, 0) \mid i=1,2, \cdots, n\} \\
& \cup\{(-i, j) \mid i, j=1,2, \cdots, n\} \\
& \cup\{(i,-j) \mid i, j=1,2, \cdots, n\} .
\end{aligned}
$$

At this point, $|\Delta(K)|=2 n^{2}+4 n+1$.
On the other hand, prove: For any connected set $K$ satisfying $|K|=2 n+1$,
$$
|\Delta(K)| \leqslant 2 n^{2}+4 n+1 .
$$

Construct a graph $G$, such that the vertices of $G$ are the points in $K$. If two points in $G$ correspond to points in $K$ with the same y-coordinate and a difference of 1 in the x-coordinate, then connect them with a red edge; if two points in $G$ correspond to points in $K$ with the same x-coordinate and a difference of 1 in the y-coordinate, then connect them with a blue edge. Since $K$ is connected, $G$ is a connected graph.

Take a spanning tree $G_{0}$ of $G$, and let the number of red and blue edges in $G_{0}$ be $e_{r}$ and $e_{b}$, respectively. Then $e_{r}+e_{b}=2 n$.

Let the red edges in $G_{0}$ be $A_{i} A_{i}^{\prime}(i=1,2, \cdots, e_{r})$, and the blue edges be $B_{i} B_{i}^{\prime}(i=1,2, \cdots, e_{b})$, and use the same notation for the corresponding edges in $K$.
Assume $A_{i}^{\prime}$ is to the right of $A_{i}$, and $B_{i}^{\prime}$ is above $B_{i}$.
Now, construct a graph $M$, such that the vertices of $M$ are the vectors formed by any two different points in $K$, some of which may be equal. Then
$$
|M|=2 \mathrm{C}_{2 n+1}^{2}=4 n^{2}+2 n \text {. }
$$
Connect the vertices in $M$ corresponding to $\overrightarrow{A_{i} A_{j}}$ and $\overrightarrow{A_{i}^{\prime} A_{j}^{\prime}}\left(i \neq j, 1 \leqslant i, j \leqslant e_{r}\right)$ with a red edge; connect the vertices in $M$ corresponding to $\overrightarrow{B_{i} B_{j}}$ and $\overrightarrow{B_{i}^{\prime} B_{j}^{\prime}}\left(i \neq j, 1 \leqslant i, j \leqslant e_{b}\right)$ with a blue edge.

For the graph $M$ thus obtained, since the $A_{i}$ are distinct and the $B_{j}$ are distinct, no two vertices in $M$ are connected by more than one edge, and $\overrightarrow{A_{i}^{\prime} A_{j}^{\prime}}$ is a leftward translation of $\overrightarrow{A_{i} A_{j}}$ by one unit. Therefore, no two vertices in $M$ are connected by both a red and a blue edge. This means $M$ is a simple graph.
$$
\begin{array}{l}
\text { If } M \text { contains a cycle: } \\
\overrightarrow{C_{1} D_{1}} \rightarrow \overrightarrow{C_{2} D_{2}} \rightarrow \cdots \rightarrow \overrightarrow{C_{k} D_{k}} \rightarrow \overrightarrow{C_{1} D_{1}} \\
\end{array}
$$
(connected by both red and blue edges), then $C_{1}$, $C_{2}, \cdots, C_{k}, C_{1}$ is a cycle in $G_{0}$, which contradicts the fact that $G_{0}$ is a tree.
Therefore, $M$ is a union of several trees.
Since two vertices in $M$ being connected by an edge means the corresponding vectors in $K$ are equal, the number of non-zero vectors in $K$ equals the number of connected components in $M$, i.e., $|M|-e(M)$ (where $e(M)$ is the number of edges in $M$). Then
$$
\begin{array}{l}
|\Delta(K)|=|M|-e(M)+1 \\
=4 n^{2}+2 n-2 \mathrm{C}_{e_{r}}^{2}-2 \mathrm{C}_{e_{b}}^{2}+1 \\
=4 n^{2}+4 n+1-\left(e_{r}^{2}+e_{b}^{2}\right) \\
\leqslant 4 n^{2}+4 n+1-\frac{1}{2}\left(e_{r}+e_{b}\right)^{2} \\
=2 n^{2}+4 n+1 .
\end{array}
$$

In summary, the maximum value sought is $2 n^{2}+4 n+1$.