Multiple choice

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

  1. $\angle RPQ$
  2. $2\angle RPQ$
  3. $3\angle RPQ$
  4. $4\angle RPQ$
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B Correct answer
Explanation

Let angle OPQ = theta. Since PR is tangent, angle OPR = 90. Angle RPQ = 90 - theta. In triangle OPQ, OP = OQ (radii), so angle OQP = angle OPQ = theta. Angle POQ = 180 - 2 * theta. This doesn't directly relate to RPQ unless we use the property that the angle between tangent and chord equals the angle in the alternate segment. Angle RPQ = angle in alternate segment. This is a standard circle theorem.

AI explanation

Since PR is tangent to the circle at P and PO is a radius, angle OPR is 90 degrees. In triangle POQ, the sides OP and OQ are both radii, making it an isosceles triangle where angle OPQ equals angle OQP. Because angles OPR and OPQ are supplementary, angle OQP is 90 degrees minus angle RPQ, meaning angle POQ equals 180 degrees minus 2 times 90 degrees minus angle RPQ, which simplifies to 2 times angle RPQ.

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