Question

In the given figure, AOC is a diameter of a circle with centre O and arc $AXB=\frac{1}{2}$ arc BYC. Find ∠BOC.

Answer 1

We have:
$\mathrm{arc}AXB=\frac{1}{2}\mathrm{arc}BYC$

$\Rightarrow \angle \mathrm{AOB}=\frac{1}{2}\angle \mathrm{BOC}$

But ∠AOB + ∠BOC = 180°

$\Rightarrow \frac{1}{2}\angle \mathrm{BOC}+\angle \mathrm{BOC}=180°$

$\Rightarrow \frac{3}{2}\angle \mathrm{BOC}=180°$

$\Rightarrow \angle \mathrm{BOC}=\left(180°\times \frac{2}{3}\right)=120°$

∴ ∠BOC = 120°