In ΔABC, BE is a median meeting AC at E. If the centroid is G and BG = 6 cm, find the length of the whole median BE.

Introduction / Context:
The centroid G divides each median in the ratio 2:1, counted from the vertex. This invariant allows direct recovery of the total median length when one centroid segment is known.

Given Data / Assumptions:

• BE is a median; G is the centroid.
• BG = 6 cm (segment from vertex B to centroid).

Concept / Approach:
By definition, BG : GE = 2 : 1 along median BE. Therefore BE = BG + GE = BG + (BG/2) = 3BG/2.

Step-by-Step Solution:
BE = (3/2) * BG = (3/2) * 6 = 9 cm.

Verification / Alternative check:
Compute GE = 3 cm; 6 + 3 = 9, matching BE.

Why Other Options Are Wrong:
7, 8, 10 cm contradict the fixed 2:1 centroid division.

Common Pitfalls:
Accidentally using 1:2 instead of 2:1 from the vertex.

Final Answer:
9 cm