CDS 2026 (I) Elementary Mathematics, question 63

Topic: Geometry

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Q63

A triangle PQR is inscribed in a circle with its centre at O. A tangent PT is drawn at P such that ∠QPT=36∘\angle QPT=36^{\circ}. What is ∠POQ\angle POQ equal to ?

  1. (a) 36∘36^{\circ}
  2. (b) 54∘54^{\circ}
  3. (c) 72∘72^{\circ}Correct answer
  4. (d) 108∘108^{\circ}

Solution.

Answer: (c) 72∘72^{\circ}

  1. The radius OP is perpendicular to tangent PT.
  2. Since ∠QPT=36∘,∠OPQ=90∘−36∘=54∘\angle QPT = 36^{\circ},\angle OPQ = 90^{\circ} - 36^{\circ} = 54^{\circ}.
  3. OP=OQOP = OQ, so triangle OPQ has two angles of 54°.
  4. Therefore ∠POQ=180∘−108∘=72∘\angle POQ = 180^{\circ} - 108^{\circ} = 72^{\circ}.