Question

If a chord $AB$subtends an angle of $60°$at the centre of a circle, then angle between the tangents at $A$ and$\mathbf{}B$ is also $60°$. Write ‘True’ or ‘False’ and justify your answer.

• A. True
• B. False

Answer 1

The correct option is B False

Explanation:

In the figure, we have a circle with a center $O$ and$AB$is a chord and $\angle AOB=60°.$

$OA⏊AC$ and$OB⏊CB$ [The tangent is perpendicular to radius]

Therefore, $\angle \mathrm{OBC}=\angle \mathrm{OAC}=90°\dots \left(1\right)$

According to the angle sum property of a quadrilateral:

$$\begin{gathered} \angle OBC+\angle OAC+\angle AOB+\angle ACB=360° \\ \Rightarrow 90°+90°+60°+\angle ACB=360° \\ \Rightarrow \angle ACB=120° \end{gathered}$$

The angle between two tangents is $120°$.

Hence, the statement is False.