17. The diagram shows a pattern found on a floor tile in the cathedral in Spoleto, Umbria. A circle of radius 1 surrounds four quarter circles, also of radius 1 , which enclose a square. The pattern has four axes of symmetry. What is the side length of the square?
A $\frac{1}{\sqrt{2}}$
B $2-\sqrt{2}$
C $\frac{1}{\sqrt{3}}$
D $\frac{1}{2}$
E $\sqrt{2}-1$
Solution: $\mathbf{B}$
Let $O$ be the centre of the circle, let $P$ be one of the points where two of the quarter circles meet, let $Q$ be the centre of one of these quarter circles, and let $R$ be the vertex of the square that lies on $O Q$, as shown.
Then in the triangle $O P Q$ there is a right angle at $P$ and $O P=Q P=1$, and therefore, by Pythagoras' Theorem, $O Q=\sqrt{2}$. Since $Q R=1$, it follows that $O R=\sqrt{2}-1$. It follows that the diagonal of the square has length $2(\sqrt{2}-1)$. Using, Pythagoras' Theorem again, it follows that the side length of the square is $\frac{1}{\sqrt{2}}(2(\sqrt{2}-1))=2-\sqrt{2}$.