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 correct option is:
The line segment joining R and T is parallel to the line segment joining Q and S.

Let us analyze the geometry of a regular convex hexagon to determine the positions of the letters P, Q, R, S, T, and U at the vertices. Let the vertices of the regular hexagon be labeled sequentially as 0, 1, 2, 3, 4, and 5 in a clockwise manner.

In a regular hexagon with side length d, there are three types of line segments that can be formed between any two vertices:
1. Side segments (e.g., between adjacent vertices like 0 and 1) which have a length of d.
2. Short diagonals (e.g., between vertices separated by one vertex like 0 and 2) which have a length of ^√3d.
3. Long diagonals (e.g., between opposite vertices like 0 and 3) which have a length of 2d.

Let us evaluate the given statements step-by-step to find a valid placement of the letters on the vertices:

Statement 1: The line segment joining R and S is longer than the line segment joining P and Q.
This implies that the segment RS must be a long diagonal (length 2d) or a short diagonal (length ^√3d). Let us place R and S at opposite vertices to form a long diagonal. For example, let:
R = vertex 3
S = vertex 0

Statement 2: The line segment joining R and S is perpendicular to the line segment joining P and Q.
Since RS is the long diagonal joining vertex 3 and vertex 0 (which lies along the horizontal axis of symmetry), any segment perpendicular to it must be vertical. The short diagonals joining vertices 1 and 5, or vertices 2 and 4, are perpendicular to the horizontal diagonal 3-0.
Since the length of PQ must be shorter than RS (which is 2d), PQ can be the short diagonal of length ^√3d. Let us choose:
P = vertex 5
Q = vertex 1

Statement 3: The line segment joining R and U is parallel to the line segment joining T and Q.
The remaining two vertices to be assigned to T and U are vertex 2 and vertex 4. Let us set:
U = vertex 2
T = vertex 4
Now, the segment RU joins vertex 3 and vertex 2, which is a side of the hexagon. The segment TQ joins vertex 4 and vertex 1, which is a main diagonal of the hexagon. In a regular hexagon, a side (such as the segment joining 3 and 2) is parallel to the opposite main diagonal (such as the segment joining 4 and 1). Thus, this assignment fully satisfies Statement 3.

With this configuration, the letters are placed at the vertices as follows:
Vertex 0: S
Vertex 1: Q
Vertex 2: U
Vertex 3: R
Vertex 4: T
Vertex 5: P

Let us verify the options based on this configuration:
The segment RT joins vertex 3 and vertex 4, which is a side of the hexagon.
The segment QS joins vertex 1 and vertex 0, which is the opposite side of the hexagon.
In any regular hexagon, opposite sides are always parallel to each other. Therefore, the line segment joining R and T is parallel to the line segment joining Q and S.