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GATE 2026 CE (CE1) – Question 7

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

In the given figure, $\overline{PQ}$ is the diameter of a circle with center $O$. Two points $R$ and $S$ are chosen on the circle such that $\angle ROS = 80°$. When $\overline{PR}$ and $\overline{QS}$ are extended, they meet at $T$. The value of $\angle RTS$ is __________.

A circle with diameter PQ and centre O. Points R and S lie on the upper half of the circle, with angle ROS marked as 80°. The lines PR and QS are extended beyond R and S to meet at the point T above the circle.
  1. 40°
  2. 50°
  3. 60°
  4. 80°

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Correct answer: (B) 50°

Explanation

The angle between two secants that meet outside a circle is half the difference of the two arcs they cut off. The far arc is the semicircle PQ, which is 180°, and the near arc RS is 80°. So $\angle RTS = \frac{180° - 80°}{2} = 50°$.