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?

Let us analyze the structure of a regular convex hexagon to determine the placement of the letters P, Q, R, S, T, and U at its six vertices.

Let the vertices of the regular hexagon be numbered sequentially as 1, 2, 3, 4, 5, and 6 in a clockwise direction. The distance between any two vertices depends on the number of steps along the perimeter separating them. Specifically:
- The shortest distance between distinct vertices corresponds to adjacent vertices (e.g., 1 and 2, step distance of 1).
- A longer distance corresponds to vertices separated by one intermediate vertex (e.g., 1 and 3, step distance of 2).
- The longest possible distance (diameter of the hexagon) corresponds to opposite vertices (e.g., 1 and 4, step distance of 3).

Let us examine the given statements step-by-step to constrain the positions of the letters:

Statement 1: "The line segment joining R and S is longer than the line segment joining P and Q."
In a regular hexagon, the line segment lengths can be classified as short (adjacent vertices, length $s$), medium (vertices separated by one vertex, length $\sqrt{3}s$), or long (opposite vertices, length $2$$s$).
Since segment RS is longer than segment PQ, segment RS must be either a medium or a long diagonal. Segment PQ can only be a short or a medium segment.

Statement 2: "The line segment joining R and S is perpendicular to the line segment joining P and Q."
Let us identify which pairs of segments in a regular hexagon are perpendicular to each other:
- A medium diagonal (e.g., segment joining 1 and 3) is perpendicular to the side (adjacent segment) that is parallel to the symmetry axis of the segment (e.g., side 1-6 or 3-4, or the opposite medium diagonal, or the side connecting the other vertices). Specifically, a medium diagonal connecting vertices $i$ and $i+2$ is perpendicular to the adjacent sides $i$-$(i-1)$ and $(i+2)$-$(i+3)$, and it is also perpendicular to the opposite sides.
- Alternatively, two diagonals can be perpendicular if one is a long diagonal (e.g., 1-4) and the other is a perpendicular medium diagonal (e.g., 2-6 or 3-5).
Since RS is longer than PQ and they are perpendicular, the only geometric configuration matching this is:
- RS is a medium diagonal (length $\sqrt{3}s$) and PQ is a short segment (side of the hexagon, length $s$).
- Alternatively, RS is a long diagonal (length $2$$s$) and PQ is a medium diagonal (length $\sqrt{3}s$). Let us test the first case: RS is a medium diagonal. Let R be at vertex 1 and S be at vertex 3. The segment RS is perpendicular to the side connecting vertices 5 and 6, or 4 and 5. Let's place P and Q at vertices 5 and 6, or 4 and 5.
Let's choose the following coordinate-free vertex position assignment to satisfy these conditions:
Let R be at vertex 1.
Let S be at vertex 3. (RS is a medium diagonal).
For PQ to be perpendicular to RS and shorter than RS, PQ must be a side parallel to the axis of symmetry perpendicular to RS. The side perpendicular to the diagonal joining 1 and 3 is the side connecting vertices 4 and 5.
Thus, we assign P and Q to vertices 4 and 5. Without loss of generality, let Q be at vertex 4 and P be at vertex 5.

Now we have placed four letters:
R = 1
S = 3
Q = 4
P = 5
The remaining empty vertices are 2 and 6. The remaining letters to place are T and U.

Statement 3: "The line segment joining R and U is parallel to the line segment joining T and Q."
We have R at vertex 1 and Q at vertex 4. The empty vertices for T and U are 2 and 6.
- If we place U at vertex 2, then T must be at vertex 6. Let's check the segments RU and TQ:
RU joins vertex 1 and vertex 2 (a side of the hexagon).
TQ joins vertex 6 and vertex 4 (a medium diagonal).
A side of a hexagon (1-2) is parallel to the opposite side (4-5), not a medium diagonal. Thus, this placement does not yield parallel segments.
- If we place U at vertex 6, then T must be at vertex 2. Let's check the segments RU and TQ:
RU joins vertex 1 and vertex 6 (a side of the hexagon).
TQ joins vertex 2 and vertex 4 (a medium diagonal).
Again, a side is not parallel to a medium diagonal.
Therefore, our assumption that RS is a medium diagonal and PQ is a side requires refinement. Let's swap the orientation or try the other case: RS is a long diagonal.
Let R = 1 and S = 4 (RS is a long diagonal of length 2).
For PQ to be perpendicular to RS and shorter than RS:
The segments perpendicular to the long diagonal 1-4 are the medium diagonals 2-6 and 3-5. Since PQ must be shorter than RS, PQ must be a medium diagonal of length $\sqrt{3}$.
Let's place P and Q at vertices 2 and 6. Without loss of generality, let Q = 2 and P = 6.
The remaining vertices are 3 and 5, which must contain T and U.
We are given that the line segment joining R and U is parallel to the line segment joining T and Q.
R is at vertex 1, and Q is at vertex 2.
We need to place T and U at {3, 5} such that RU is parallel to TQ.
Let's test the two possibilities:
1. If U = 3 and T = 5:
RU is the segment joining 1 and 3 (medium diagonal).
TQ is the segment joining 5 and 2 (medium diagonal).
In a regular hexagon, the diagonal 1-3 is indeed parallel to the diagonal 5-2.
2. If U = 5 and T = 3:
RU is the segment joining 1 and 5 (medium diagonal).
TQ is the segment joining 3 and 2 (side).
A medium diagonal 1-5 is not parallel to a side 3-2.
Thus, the correct configuration of vertices is:
Vertex 1 = R
Vertex 2 = Q
Vertex 3 = U
Vertex 4 = S
Vertex 5 = T
Vertex 6 = P

Let us verify the statements with this assignment:
- RS joins 1 and 4 (long diagonal, length 2).
- PQ joins 6 and 2 (medium diagonal, length $\sqrt{3}$).
Since $2>\sqrt{3}$, RS is longer than PQ. (Statement 1 holds).
- The long diagonal 1-4 is perpendicular to the medium diagonal 2-6. (Statement 2 holds).
- RU joins 1 and 3, and TQ joins 5 and 2. These two medium diagonals are parallel. (Statement 3 holds).

Now we evaluate the given options to find which statement is CORRECT:
1. "The line segment joining T and Q is parallel to the line joining P and U."
TQ joins 5 and 2. PU joins 6 and 3. In a regular hexagon, 5-2 is a medium diagonal, and 6-3 is a long diagonal. They are not parallel (they intersect). Thus, this option is incorrect.
2. "The line segment joining Q and S is perpendicular to the line segment joining R and P."
QS joins 2 and 4 (medium diagonal). RP joins 1 and 6 (side). They are not perpendicular. Thus, this option is incorrect.
3. "The line segment joining R and T is parallel to the line segment joining Q and S."
RT joins 1 and 5 (medium diagonal). QS joins 2 and 4 (medium diagonal). In a regular hexagon, the medium diagonal connecting vertices 1 and 5 is parallel to the medium diagonal connecting vertices 2 and 4. Thus, this option is correct.
4. "The line segment joining R and P is perpendicular to the line segment joining U and Q."
RP joins 1 and 6 (side). UQ joins 3 and 2 (side). Side 1-6 and side 2-3 are not perpendicular (they intersect at an angle of 60 degrees). Thus, this option is incorrect.

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