A2. The center of a circle with radius 1 coincides with the intersection of the diagonals of a square.

The circle and the square have the same

![](https://cdn.mathpix.com/cropped/2024_04_17_9dc9851d634283e8a951g-1.jpg?height=340&width=323&top_left_y=1640&top_left_x=1546)
area. The circle intersects a side of the square at points $A$ and $B$.

How long is the line segment $A B$?
A2. The area of the circle is equal to $\pi$. The area of the square is thus also $\pi$ and the side of the square is $\sqrt{\pi}$.

$M C=\frac{1}{2} \sqrt{\pi}$ and $A M=1$.

Using Pythagoras: $\left(\frac{1}{2} A B\right)^{2}=A C^{2}=1^{2}-\left(\frac{1}{2} \sqrt{\pi}\right)^{2}=1-\frac{1}{4} \pi$.

![](https://cdn.mathpix.com/cropped/2024_04_17_bf0efe45fea0def43891g-1.jpg?height=268&width=257&top_left_y=563&top_left_x=1528)

So $A B^{2}=4-\pi$ and $A B=\sqrt{4-\pi}$.