17. The answer is $\mathbf{( C )}$.

$Q^{\prime}$ is a square inscribed in the circle and can be drawn indifferently with sides parallel to those of $Q$ or with vertices at the points of contact between $Q$ and the circle.

By drawing the diagonals of $Q^{\prime}$, the square $Q$ is divided into 8 congruent isosceles right triangles, of which four are also part of $Q^{\prime}$. The ratio between the two areas is $8 / 4=2$.

![](https://cdn.mathpix.com/cropped/2024_04_17_4701f996ee50400fb9c5g-4.jpg?height=317&width=320&top_left_y=1509&top_left_x=1576)
17) In the adjacent figure, let's call $Q$ the square circumscribed around the circle and $Q^{\prime}$ the square inscribed in the circle. What is the ratio between the area of $Q$ and that of $Q^{\prime}$?
(A) $\frac{\sqrt{2}}{2}$,
(B) $\sqrt{2}$,
(C) 2,
(D) $2 \sqrt{2}$,
(E) 4.

![](https://cdn.mathpix.com/cropped/2024_04_17_84f5f6e83b5066604abfg-2.jpg?height=257&width=258&top_left_y=916&top_left_x=1031)