Question

In the figure below, AB and AC are chords of the circle and OP and OQ are radii parallel to them:

Find the ratio of $\angle BOC$ and $\angle POQ$ .

Answer 1

Let $\angle ABO=x$ and $\angle ACO=y$

then, $\angle BAC=x+y$

$\angle BOC=2(x+y)$ (1 mark)

Since, AB is parallel to OP

So, $\angle BOP=x$ (alternate interior angles)

Similarly, AC is parallel to OQ

So, $\angle QOC=y$ (alternate interior angles) (1 mark)

Now,

$\angle BOC=2(x+y)$

So, $\angle POQ=\angle BOC-(x+y)$

$=2(x+y)-(x+y)$

$=(x+y)$

So, $\frac{\angle BOC}{\angle POQ}=\frac{2(x+y)}{(x+y)}=2$ (1 mark)