In a triangle, the distances from the centroid to the three vertices are 4 cm, 6 cm and 8 cm respectively. Using properties of medians and centroid, what is the length of the smallest median (in centimetres)?

Introduction / Context:
This triangle problem uses the fact that the centroid divides each median in a fixed ratio. If you know the distance from the centroid to each vertex, you can reconstruct the full median lengths and then identify the shortest one. This is a conceptual geometry question that combines knowledge of triangle centres with simple proportional reasoning.

Given Data / Assumptions:

• G is the centroid of the triangle.
• Distances from G to the three vertices are 4 cm, 6 cm, and 8 cm.
• We must find the length of the smallest median of the triangle.
• All medians are assumed to be drawn correctly from vertices to midpoints of opposite sides.

Concept / Approach:
A key property of the centroid is: The centroid divides each median in the ratio 2 : 1 from vertex to midpoint. That is, the distance from a vertex to the centroid is 2/3 of the full median length. So for a median of length m, the segment from vertex to centroid is (2/3)m, and from centroid to midpoint is (1/3)m. We can invert this to find m if we know the vertex–centroid distance.

Step-by-Step Solution:
Let the three medians have lengths m1, m2, and m3. Given vertex–centroid distances are 4 cm, 6 cm, and 8 cm. For each median, vertex–centroid distance = (2/3) * median length. So m1 = (3/2) * 4 = 6 cm. m2 = (3/2) * 6 = 9 cm. m3 = (3/2) * 8 = 12 cm. The smallest median is the one with length 6 cm.

Verification / Alternative check:
We can verify the ratios quickly: for the median of length 6 cm, 2/3 * 6 = 4 cm, matching one given distance. Similarly, 2/3 * 9 = 6 and 2/3 * 12 = 8. This confirms that we correctly reconstructed all median lengths from the centroid distances.

Why Other Options Are Wrong:
8, 7, and 5 do not match any of the median lengths deduced. 8 and 9 might be confused with the given centroid distances or other medians, but the smallest median computed is 6 cm, not 5 cm or 7 cm or 8 cm.

Common Pitfalls:
It is easy to invert the 2 : 1 ratio incorrectly and use 1/2 or 2 instead of 3/2. Some students also assume the centroid divides the median in half, which is incorrect. Always remember the precise ratio: vertex to centroid is two-thirds of the median, centroid to midpoint is one-third.

Final Answer:
The smallest median of the triangle has length 6 cm.