16. Let $O$ be the centre of the circumcircle of $\triangle A B C, P$ and $Q$ the midpoints of $A O$ and $B C$, respectively. Suppose $\angle C B A=4 \angle O P Q$ and $\angle A C B=6 \angle O P Q$. Find the size of $\angle O P Q$ in degrees.
16. Ans: 12
Let $R$ be the midpoint of $O C$. Then $P O=R O=R Q$. Let $\angle O P Q=x$. Then $\angle P O R=2 \angle B=8 x$ and $\angle B O A=2 \angle C=12 x$. Also, $\angle R O Q=\angle A=180^{\circ}-10 x$. It follows that $\angle P Q O=180^{\circ}-x-8 x-\left(180^{\circ}-10 x\right)=x$ and therefore $P O=O Q$. Thus $\triangle O Q R$ is equilateral, so $\angle R O Q=180^{\circ}-10 x=60^{\circ}$ and $x=12^{\circ}$.