24. The diagram shows a square $P Q R S$. The arc $Q S$ is a quarter circle. The point $U$ is the midpoint of $Q R$ and the point $T$ lies on $S R$. The line $T U$ is a tangent to the arc $Q S$.
What is the ratio of the length of $T R$ to the length of $U R$ ?
A $3: 2$
B $4: 3$
C $5: 4$
D $7: 6$
E $9: 8$
Solution
B
It is convenient, because $U$ is the midpoint of $Q R$, to choose units so that the square has sides of length 2 . It follows that each of $U Q$ and $U R$ has length 1 .
Suppose that $T R$ has length $x$. Then the length of $T S$ is $2-x$.
Let $V$ be the point where the line $T U$ is tangent to the arc.
Because the two tangents from a point to a circle have the same length, $U V=U Q=1$ and $V T=T S=2-x$. It follows that $U T=U V+V T=1+(2-x)=3-x$.
In the right-angled triangle $U R T$ the hypotenuse has length $3-x$, and the other two sides have lengths 1 and $x$. Therefore, by Pythagoras' theorem applied to this triangle,
$$
1^{2}+x^{2}=(3-x)^{2} \text {. }
$$

It follows that
$$
1+x^{2}=9-6 x+x^{2},
$$
and hence
$$
6 x=8
$$

We deduce that $x=\frac{4}{3}$.
Therefore $T R: U R=\frac{4}{3}: 1=4: 3$.