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GATE DA 2026 Q10: Area of a dart-shaped region inside a circle

Question

In the given figure, PP, QQ, and RR are three points on a circle of radius 10 cm with OO as its center, PQ‾=RQ‾\overline{PQ} = \overline{RQ}, and ∠PQR=45∘\angle PQR = 45^\circ. The figure is representative.

The area of the shaded region PQROPQRO is ______________ cm².

Circle with centre O; points P, Q, R on the circle with angle PQR of 45 degrees; the region P-Q-R-O is shaded
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Wrong answer costs 2/3 mark in GATE.
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Answer: C (official GATE 2026 final answer key)

Concept used: Inscribed angle theorem and the triangle area 12absin⁡C\tfrac{1}{2}ab\sin C.

First step: ∠POR=2×45∘=90∘\angle POR = 2 \times 45^\circ = 90^\circ, and by symmetry ∠POQ=∠QOR=135∘\angle POQ = \angle QOR = 135^\circ; the region is triangles OPQOPQ and OQROQR.

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Source: official GATE 2026 DA question paper and final answer key, organised by IIT Guwahati. Question paper · Answer key

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