24. In square $R S T U$ a quarter-circle arc with centre $S$ is drawn from $T$ to $R$. A point $P$ on this arc is 1 unit from $T U$ and 8 units from $R U$. What is the length of the side of square RSTU?
A 9
B 10
C 11
D 12
E 13
24. E In the diagram $V$ is the point where the perpendicular from $P$ meets TS. Let the side of the square RSTU be $x$. So the radius of the arc from $R$ to $T$ is $x$. Therefore $S P$ has length $x, P V$ has length $x-8$ and $V S$ has length $x-1$. Applying Pythagoras' Theorem to triangle $P V S$ :
$$
(x-8)^{2}+(x-1)^{2}=x^{2} \text {. So } x^{2}-16 x+64+x^{2}-2 x+1=x^{2} \text {. }
$$

Therefore $x^{2}-18 x+65=0$, so $(x-5)(x-13)=0$.
Hence $x=5$ or $x=13$, but $x>8$ so the length of the side of the square RSTU is 13 .