Question

If $\overline{P}Q$ is a chord of a circle with centre $O$ and $PR$ is a tangent to the circle at $P,$ then $\angle POQ=$

• A. $4$ $\angle RPQ$
• B. $3$ $\angle RPQ$
• C. $2$ $\angle RPQ$
• D. $\angle RPQ$

Answer 1

The correct option is A $4$ $\angle RPQ$

Let $\angle RPQ=X$

By tangent rule we know that $\angle OPR=90$

$\to \angle OPR=\angle OPQ+\angle RPQ={90}^o$ ...(1)

Now angle $\angle OPQ=\angle OQP$ (Angles opposide to equal sides are equal)

$\to$ Sum of all angles of triangle = ${180}^o$

$\to \angle OPQ+\angle OQP+\angle POQ={180}^o\Rightarrow \angle POQ=180-2\angle OPQ$

$\therefore {180}^o-2(90-\angle RPQ)=\angle POQ$ (from (1))

$\therefore \angle POQ=2\angle RPQ$