Question

For the question given below,four alternative choices have been provided of which only one is correct. You have to select the correct choice:

In Fig. 1, O is the centre of the circle and ∠OBA = 60°. Then ∠ACB equals:

(a) 60°

(b) 45°

(c) 30°

(d) 90°

Answer 1

We have, ∠OBA = 60°

In $∆$AOB,

As, OA = OB (Radii)

So, ∠OAB = ∠OBA (Angles opposite to equal sides are equal)

Therefore, ∠OAB = 60°

Also, using angle sum property of triangle

$$\begin{gathered} \angle \mathrm{AOB}=180°-\left(\angle \mathrm{OBA}+\angle \mathrm{OAB}\right) \\ =180°-\left(60°+60°\right) \\ =180°-120° \\ =60° \end{gathered}$$

Now,

$$\begin{gathered} \angle \mathrm{AOB}=2\angle \mathrm{ACB}\left(\mathrm{Angle}\mathrm{subtended}\mathrm{at}\mathrm{the}\mathrm{center}\mathrm{by}\mathrm{a}\mathrm{chord}\mathrm{is}\mathrm{twice}\mathrm{the}\mathrm{angle}\mathrm{subtended}\mathrm{by}\mathrm{the}\mathrm{same}\mathrm{chord}\mathrm{at}\mathrm{the}\mathrm{remaining}\mathrm{part}\mathrm{of}\mathrm{the}\mathrm{circle}\right) \\ \Rightarrow \angle \mathrm{ACB}=\frac{\angle \mathrm{AOB}}{2} \\ \Rightarrow \angle \mathrm{ACB}=\frac{60°}{2} \\ \therefore \angle \mathrm{ACB}=30° \end{gathered}$$

Hence, the correct choice is option (c).