23. A Saxon silver penny, from the reign of Ethelbert II in the eighth century, was sold in 2014 for $£ 78000$. A design on the coin depicts a circle surrounded by four equal arcs, each a quarter of a circle, as shown. The width of the design is $2 \mathrm{~cm}$.
What is the radius of the small circle, in centimetres?

A $\frac{1}{2}$
B $2-\sqrt{2}$
C $\frac{1}{2} \sqrt{2}$
D $5-3 \sqrt{2}$
E $2 \sqrt{2}-2$
23. B The diagram shows part of the figure, to which have been added $A$ and $C$, centres of two of the quarter-circle arcs, $B$ and $D$, points of intersection of two arcs, and $E$, the centre of the small circle. In cm, the radii of each arc and the small circle are $R$ and $r$ respectively. Firstly, note that $\angle B C D$ is a right angle as arc $B D$ is a quarter of a circle. Therefore, by Pythagoras' Theorem $R^{2}+R^{2}=2^{2}$ so $R=\sqrt{2}$. Consider triangle $A C E$ : from the symmetry of the figure we deduce that $\angle A E C=\frac{1}{4} \times 360^{\circ}=90^{\circ}$. So, by Pythagoras' Theorem $(R+r)^{2}+(R+r)^{2}=(2 R)^{2}=4 R^{2}$. Therefore $(R+r)^{2}=2 R^{2}=2 \times 2=4$. Hence $R+r=2$, so $r=2-R=2-\sqrt{2}$.