18. In the figure below, $A B$ and $C D$ are parallel chords of a circle with centre $O$ and radius $r \mathrm{~cm}$. It is given that $A B=46 \mathrm{~cm}, C D=18 \mathrm{~cm}$ and $\angle A O B=3 \times \angle C O D$. Find the value of $r$.
18. Answer: 27

Let $M$ and $N$ be the midpoints of $A B$ and $C D$ respectively and let $\angle C O N=x$.
Then $\angle A O N=3 x$ and
$$
\frac{23}{9}=\frac{A M}{C N}=\frac{r \sin 3 x}{r \sin x}=\frac{3 \sin x-4 \sin ^{3} x}{\sin x}=3-4 \sin ^{2} x \text {. }
$$

Thus $\sin ^{2} x=\frac{1}{4}\left(3-\frac{23}{9}\right)=\frac{1}{9}$, and so $\sin x=\frac{1}{3}$.
Hence $r=\frac{C N}{\sin x}=27$.