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GATE 2022 EE – Question 9

General Aptitude · Analytical Aptitude: Logical Reasoning and Deduction · 2 marks · Multiple choice

The letters P, Q, R, S, T and U are to be placed one per vertex on a regular convex hexagon, but not necessarily in the same order.

Consider the following statements:

Based on the above statements, which one of the following options is CORRECT?

  1. The line segment joining R and T is parallel to the line segment joining Q and S
  2. The line segment joining T and Q is parallel to the line joining P and U
  3. The line segment joining R and P is perpendicular to the line segment joining U and Q
  4. The line segment joining Q and S is perpendicular to the line segment joining R and P

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Show answer and explanation

Correct answer: (A) The line segment joining R and T is parallel to the line segment joining Q and S

Explanation

In a regular hexagon the only perpendicular pairs made of four distinct vertices are a long diagonal and a short diagonal, so RS is a long diagonal and PQ a short diagonal perpendicular to it. Label the vertices $1,\dots,6$ with $R=1$, $S=4$, and $P,Q=5,3$. Then RU $\parallel$ TQ forces $U=2$, $T=6$, $Q=3$ (the segments $1$–$2$ and $6$–$3$ are parallel). Then RT is the segment $1$–$6$ and QS is the segment $3$–$4$, which are parallel, so A holds. The other options fail for this labelling.