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GATE 2025 EE – Question 10

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

In the triangle PQR, the lengths of PT and TR are in the ratio of 3:2.

ST is parallel to QR. Two semicircles are drawn with PS and PQ as diameters, as shown in the figure.

Which one of the following statements is true about the shaded area PQS?

Note: The figure shown is representative.

Diagram for GATE 2025 EE question 10
  1. The shaded area is $\dfrac{16}{9}$ times the area of the semicircle with the diameter PS.
  2. The shaded area is equal to the area of the semicircle with the diameter PS.
  3. The shaded area is $\dfrac{14}{9}$ times the area of the semicircle with the diameter PS.
  4. The shaded area is $\dfrac{14}{25}$ times the area of the semicircle with the diameter PQ.

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Correct answer: (A) The shaded area is $\dfrac{16}{9}$ times the area of the semicircle with the diameter PS.

Explanation

Since $ST\parallel QR$, $\dfrac{PS}{PQ}=\dfrac{PT}{PR}=\dfrac35$. The shaded region is the large semicircle (diameter PQ) minus the small one (diameter PS). Semicircle areas are proportional to the square of the diameter, so with $PS=3$ and $PQ=5$: shaded $=\dfrac{5^2-3^2}{3^2}=\dfrac{16}{9}$ times the semicircle on PS.