Question:
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?
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?
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?
Show Hint
In a regular convex hexagon, opposite sides and diagonals often exhibit parallel and perpendicular relationships due to symmetry. Always consider the geometry of the shape while analyzing such problems.
Updated On: Nov 27, 2025
- The line segment joining R and T is parallel to the line segment joining Q and S
- The line segment joining T and Q is parallel to the line joining P and U
- The line segment joining R and P is perpendicular to the line segment joining U and Q
- The line segment joining Q and S is perpendicular to the line segment joining R and P
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Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Understanding the regular convex hexagon.
In a regular convex hexagon, the internal angles are all equal, and the sides are of equal length. Moreover, the diagonals connecting non-adjacent vertices are symmetrical.
Step 2: Analyzing the given statements.
- The line segment joining R and S is longer than the line segment joining P and Q, indicating that R and S are connected by a longer diagonal, while P and Q are connected by a shorter one.
- The line segment joining R and S is perpendicular to the line segment joining P and Q, meaning that the diagonals R-S and P-Q are orthogonal to each other.
- The line segment joining R and U is parallel to the line segment joining T and Q, suggesting that certain diagonals are aligned and parallel to each other.
Step 3: Analyzing the options.
- (A) The line segment joining R and T is parallel to the line segment joining Q and S. This is the correct option because R and T are positioned in such a way that the diagonal joining them is parallel to the diagonal joining Q and S, according to the symmetry of the hexagon.
- (B) The line segment joining T and Q is parallel to the line joining P and U. This is incorrect based on the hexagonal geometry.
- (C) The line segment joining R and P is perpendicular to the line segment joining U and Q. This is incorrect because the diagonals R-P and U-Q are not orthogonal.
- (D) The line segment joining Q and S is perpendicular to the line segment joining R and P. This is incorrect because there is no perpendicular relationship between Q-S and R-P in the given configuration.
Step 4: Final Answer.
The correct answer is (A) because the diagonals R-T and Q-S are parallel in a regular convex hexagon.
In a regular convex hexagon, the internal angles are all equal, and the sides are of equal length. Moreover, the diagonals connecting non-adjacent vertices are symmetrical.
Step 2: Analyzing the given statements.
- The line segment joining R and S is longer than the line segment joining P and Q, indicating that R and S are connected by a longer diagonal, while P and Q are connected by a shorter one.
- The line segment joining R and S is perpendicular to the line segment joining P and Q, meaning that the diagonals R-S and P-Q are orthogonal to each other.
- The line segment joining R and U is parallel to the line segment joining T and Q, suggesting that certain diagonals are aligned and parallel to each other.
Step 3: Analyzing the options.
- (A) The line segment joining R and T is parallel to the line segment joining Q and S. This is the correct option because R and T are positioned in such a way that the diagonal joining them is parallel to the diagonal joining Q and S, according to the symmetry of the hexagon.
- (B) The line segment joining T and Q is parallel to the line joining P and U. This is incorrect based on the hexagonal geometry.
- (C) The line segment joining R and P is perpendicular to the line segment joining U and Q. This is incorrect because the diagonals R-P and U-Q are not orthogonal.
- (D) The line segment joining Q and S is perpendicular to the line segment joining R and P. This is incorrect because there is no perpendicular relationship between Q-S and R-P in the given configuration.
Step 4: Final Answer.
The correct answer is (A) because the diagonals R-T and Q-S are parallel in a regular convex hexagon.
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