In the given figure, P, Q, and R are three points on a circle of radius 10 cm with O as its center, PQ = RQ, and PQR = 45°. The figure is representative. The area of the shaded region PQRO is cm2.

GATE 2026 DA – Question 10

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 2 marks · Multiple choice

In the given figure, $P$, $Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45°$. The figure is representative. The area of the shaded region $PQRO$ is ______________ cm$^2$.

A circle with centre O and points P and R on its lower left and lower right, and Q at the top. The shaded region is the four-sided arrowhead Q-P-O-R, with the 45° angle marked at Q.

50
$25\sqrt{2}$
$50\sqrt{2}$
100

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Correct answer: (C) $50\sqrt{2}$

Explanation

The angle at Q is an inscribed angle on the chord PR, so the central angle is $\angle POR = 2 \times 45^\circ = 90^\circ$. Then $PR = 10\sqrt{2}$ and the distance from O to PR is $\frac{10}{\sqrt{2}} = 5\sqrt{2}$. Since $PQ = RQ$, Q is on the perpendicular bisector of PR, on the far side of O, at a distance $10 + 5\sqrt{2}$ from the chord. The shaded arrowhead is triangle PQR minus triangle POR: $\frac{1}{2}(10\sqrt{2})(10 + 5\sqrt{2}) - \frac{1}{2}(10\sqrt{2})(5\sqrt{2}) = \frac{1}{2}(10\sqrt{2})(10) = 50\sqrt{2}$ cm$^2$.