Example 6 As shown in Figure $9, A, B, C$ are three villages located on an east-west oriented highway, $A B=$ $2 \text{ km}, B C=3 \text{ km}$. To the due north of village $B$ is village $D$, and it is measured that $\angle A D C$ $=45^{\circ}$. Now the $\triangle A D C$ area is planned as a development zone, except for a pond of $4 \text{ km}^{2}$, the rest is designated for construction or green space. Try to find the area of the construction and green space in this development zone?
(1995, Beijing Junior High School Mathematics Competition Final)
Analysis: The problem is clearly to find the length of $BD$, but there is no effective connection between the problem and the given conditions. Since $\angle ADC = 45^{\circ}$, considering its relationship with the right angle, we think of performing an axial symmetry transformation on $\triangle DAB$ and $\triangle DCB$ (as shown in Figure 9), then $\angle EDF = 90^{\circ}$. Thus, the incomplete figure is completed into a square $DEGF$.
Let $DB = x$. Then
$$
\begin{array}{l}
AG = EG - EA = DB - BA = x - 2, \\
CG = GF - CF = DB - BC = x - 3 .
\end{array}
$$

By the Pythagorean theorem, we have
$$
(x-2)^{2}+(x-3)^{2}=25 \text{. }
$$

Solving for $x$ gives $x = 6$ or -1.
Among these, $x = -1$ does not meet the problem's requirements and is discarded, so $x = 6$. Therefore, the area is $11 \text{ km}^2$.