4. Let the rectangle $A B C D$ have vertices $A(-2,-1), B(1,-1), C(1,3), D(-2,3)$. Calculate the coordinates of the center $S$ and the radius $R$ of the circle circumscribed around the rectangle. Draw the diagram.
4. The center $S$ of the circle circumscribed around the rectangle is also the midpoint of the diagonal $A C$, so $S\left(\frac{x_{A}+x_{C}}{2}, \frac{y_{A}+y_{C}}{2}\right)$ or $S\left(-\frac{1}{2}, 1\right)$. From the sketch, it is clear that the radius $R$ is equal to half the length of the diagonal $A C$. Since $|A C|=$ $\sqrt{\left(x_{C}-x_{A}\right)^{2}+\left(y_{C}-y_{A}\right)^{2}}=\sqrt{9+16}=5$, it follows that $R=\frac{5}{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-07.jpg?height=663&width=671&top_left_y=1159&top_left_x=1132)

Sketch

Calculated coordinates of the center: $S\left(-\frac{1}{2}, 1\right)$ 1 point 1 point

Conclusion $R=\frac{1}{2} \cdot d(A, C)$ or $R=\frac{1}{2} \cdot|A C|$.

1 point

Writing or using the distance formula between points......................................................................................

Correctly substituting the data into the formula ........................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_40efadbcfacda9b65264g-07.jpg?height=105&width=1642&top_left_y=2132&top_left_x=204)