3. In a rectangular coordinate system in the plane, a point $S(2,3)$ and a square $A B C D$ are given, with three known vertices $A(3,1), C(-2,6)$, and $D(-2,1)$. The square $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is the central symmetric image of the square $A B C D$ with respect to the point $S$. Determine the coordinates of the points $B, A^{\prime}, B^{\prime}, C^{\prime}$, and $D^{\prime}$, as well as the area of the figure that is the intersection of the squares $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$. The length of the unit segment is $1 \mathrm{~cm}$.
3. In a rectangular coordinate system in the plane, a point $S(2,3)$ and a square $A B C D$ are given, with three known vertices $A(3,1), C(-2,6)$, and $D(-2,1)$. The square $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is the central symmetric image of the square $A B C D$ with respect to the point $S$. Determine the coordinates of the points $B, A^{\prime}, B^{\prime}, C^{\prime}$, and $D^{\prime}$, as well as the area of the figure that is the intersection of the squares $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$. The length of the unit segment is $1 \mathrm{~cm}$.

## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_0b00c6f52e60bdba44a4g-13.jpg?height=723&width=808&top_left_y=1620&top_left_x=241)[^0]