In triangle ABC, medians BE and CF intersect at point O, the centroid. Points P and Q are the midpoints of segments BO and CO, respectively. If the length of segment PQ is 3 cm, what is the length of segment FE (in centimetres) joining the midpoints of sides AB and AC?

Introduction / Context:
This geometry question tests understanding of medians, centroids, midpoints, and relationships between segments inside a triangle. It is a typical conceptual problem where algebraic or coordinate geometry reasoning shows that certain segments in a triangle can have equal lengths, independent of the triangle size.

Given Data / Assumptions:
- Triangle ABC has medians BE and CF drawn to sides AC and AB, respectively.
- These medians intersect at O, the centroid of triangle ABC.
- P is the midpoint of segment BO.
- Q is the midpoint of segment CO.
- Segment FE joins the midpoints of sides AB and AC.
- Length PQ = 3 cm, and we must find FE.

Concept / Approach:
There are two key ideas:
- The segment joining midpoints of two sides of a triangle (here FE) is called a mid segment and is parallel to the third side, with length equal to half of that third side.
- When medians and their midpoints are considered, the small quadrilateral PEOF is homothetic to triangle ABC, and segments PQ and FE turn out to be equal in length by vector or coordinate geometry arguments.

Step-by-Step Solution:
Step 1: Consider triangle ABC in a coordinate system for clarity. Let A, B, and C be arbitrary noncollinear points. Step 2: E is the midpoint of AC, and F is the midpoint of AB. Step 3: Medians BE and CF intersect at centroid O, which is the average of the vertices, so O = (A + B + C) / 3 in vector form. Step 4: P is the midpoint of BO, and Q is the midpoint of CO, so vectors for P and Q become averages of B with O and C with O, respectively. Step 5: Using vector algebra, one can show that the vector from F to E is equal to the vector from P to Q. Hence FE and PQ are parallel and have the same length. Step 6: Therefore, if PQ = 3 cm, then FE must also be 3 cm.

Verification / Alternative Check:
Another approach is to choose a specific simple triangle, for example with coordinates A(0,0), B(2,0), and C(0,2). When the points E, F, O, P, and Q are constructed and distances are calculated, FE and PQ both come out equal. Because the result is independent of the specific coordinates (due to similarity and centroid properties), FE = PQ in every such triangle, confirming FE = 3 cm here.

Why Other Options Are Wrong:
Option b, 6 cm, would imply FE is twice PQ, which does not match the internal similarity relations created by medians and midpoints.
Option c, 9 cm, is three times PQ and has no geometric justification in this configuration.
Option d, 12 cm, is even further from the logical length and would make FE much larger than any reasonable side length if PQ is only 3 cm.

Common Pitfalls:
Many learners try to relate FE directly to side BC without considering how P and Q are constructed from BO and CO. Others forget that centroids divide medians in a 2:1 ratio and that midpoints of segments from vertices to centroid create additional similar triangles inside the original triangle. Drawing a neat diagram and, if comfortable, assigning coordinates makes such problems much easier to handle.

Final Answer:
The length of FE is 3 cm.