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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: - 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 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

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

Solution

In a regular hexagon, the length and direction of a segment depend on how many vertices apart its endpoints are. The given conditions force specific placements of the letters so that the only consistent relation among the options is that RT is parallel to QS. Hence that option is correct.

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