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The letters $\text{P, Q, R, S, T}$ 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:

  • The line segment joining $R$ and $S$ is longer than the line segment joining $P$ and $Q$.
  • The line segment joining $R$ and $S$ is perpendicular to the line segment joining $P$ and $Q$.
  • The line segment joining $R$ and $U$ is parallel to the line segment joining $T$ and $Q$.

Based on the above statements, which one of the following options is $\text{CORRECT}$?

  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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Given that,  the following statements:

  • The line segment joining $\text{R}$ and $\text{S}$ is longer than the line segment joining $\text{P}$ and $\text{Q}$.
  • The line segment joining $\text{R}$ and $\text{S}$ is perpendicular to the line segment joining $\text{P}$ and $\text{Q}$.
  • The line segment joining $\text{R}$ and $\text{U}$ is parallel to the line segment joining $\text{T}$ and $\text{Q}$.

Based on the above statements, we can draw the ${\color{Blue}{\text{regular convex hexagon.}}}$

Now, we can see all the options. 

  1. The line segment joining $\text{R}$ and $\text{T}$ is parallel to the line segment joining $\text{Q}$ and $\text{S}{\color{Green}{-\text{True}.}}$
  2. The line segment joining $\text{T}$ and $\text{Q}$ is parallel to the line joining $\text{P}$ and $\text{U}{\color{Red}{-\text{False}.}}$
  3. The line segment joining $\text{R}$ and $\text{P}$ is perpendicular to the line segment joining $\text{U}$ and $\text{Q}{\color{Red}{-\text{False}.}}$
  4. The line segment joining $\text{Q}$ and $\text{S}$ is perpendicular to the line segment joining $\text{R}$ and $\text{P}{\color{Red}{-\text{False}.}}$

Correct Answer $:\text{A}$

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