7. Two vertices of a square with an area of $256 \mathrm{~cm}^{2}$ lie on a circle, while the other two vertices lie on a tangent to this circle. Find the radius of the circle.
Solution: $S_{A B C D}=256 \, \text{cm}^{2} \Rightarrow a=16 \, \text{cm}=x$,

$E F=2 R-x, F O=R-E F=R-(2 R-x)=x-R$,

$\triangle F C O: F O^{2}=R^{2}-F C^{2} \Rightarrow(x-R)^{2}=R^{2}-8^{2}$,

$x^{2}-2 R x+R^{2}=R^{2}-8^{2}, 2 R x=16^{2}+8^{2} \Rightarrow R=\frac{16^{2}+8^{2}}{2 \cdot 16}=10$. Answer:

$R=10$.

![](https://cdn.mathpix.com/cropped/2024_05_06_8eeeb8d6ace591217257g-15.jpg?height=351&width=374&top_left_y=978&top_left_x=1618)