Question

If PQ is a chord of a circle whose centre is 0 and PR is the tangent to the circle at the point P, then \[\angle POQ\] is equal to
A. \[\angle RPQ\]
B. \[2\angle RPQ\]
C. \[3\angle RPQ\]
D. \[4\angle RPQ\]

Answer 1

Hint: In this problem, we have to find the angle which is equivalent to \[\angle POQ\]. We know that the given data is PQ is a chord of a circle, whose centre is 0 and PR is the tangent to the circle at the point P. We can first draw the diagram from the given data, we can then use the geometric formulas to find the angle equivalent to the given angle.

Complete step by step solution:
We know that, we are given PQ is a chord of a circle whose centre is 0 and PR is the tangent to the circle at the point P.
We have to find the angle equivalent to \[\angle POQ\].
We can first draw the diagram from the given data.

We can now assume that, \[\angle OPQ=x\].
We can see that from the diagram, OP = OQ = radius.
We also know that the sum of the three sides of a triangle is \[{{180}^{\circ }}\],
\[\Rightarrow \angle OPQ+\angle OQP+\angle POQ={{180}^{\circ }}\]
 We can now write it as,
\[\Rightarrow \angle POQ={{180}^{\circ }}-2x\]……. (1)
We can also see that, radius is perpendicular to the tangent,
\[\Rightarrow \angle OPR={{90}^{\circ }}\] ………. (2)
We can now write from the diagram,
\[\Rightarrow \angle RPQ=\angle OPR-\angle OPQ\]
We can now substitute (1) and (2) in the above step, we get
\[\Rightarrow \angle RPQ={{90}^{\circ }}-x\]
Now from (1), we can write the above step as,
\[\begin{align}
  & \Rightarrow \angle RPQ=\dfrac{\angle POQ}{2} \\
 & \Rightarrow 2\angle RPQ=\angle POQ \\
\end{align}\]
Therefore, the angle equivalent to the angle \[\angle POQ\] is option B. \[2\angle RPQ\].

So, the correct answer is “Option B”.

Note: We should remember that the sum of three sides of a triangle is equal to \[{{180}^{\circ }}\]. We should concentrate on the diagram part while drawing the chord and the tangent and while marking the parts given in the question part. We should also substitute the correct data step by step to get the answer.