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

Introduction / Context:
This problem uses a key property of the centroid of a triangle. The centroid divides each median in a fixed ratio, so knowing the distances from the centroid to the vertices allows you to deduce the full lengths of the medians. The question then asks for the smallest median. This blends knowledge of triangle centres with basic proportional reasoning and is a classic style of question in geometry sections of aptitude tests.

Given Data / Assumptions:

• There is a triangle with some unknown side lengths.
• The centroid is the point where all three medians intersect.
• The distances from the centroid to the three vertices are 4 cm, 6 cm, and 8 cm.
• Each such distance is the longer segment of a median, measured from the vertex to the centroid.
• We must find the length of the smallest full median of the triangle.

Concept / Approach:
The centroid divides each median in the ratio 2 : 1, counting from the vertex to the mid point of the opposite side. This means that the segment from the vertex to the centroid is two thirds of the entire median, and the remaining one third lies between the centroid and the mid point of the opposite side. If the distance from the centroid to a vertex is d, then the full median length m satisfies d = (2 / 3) * m, so m = (3 / 2) * d. We can apply this relation to each of the three given distances and then compare the resulting medians.

Step-by-Step Solution:
Let the distances from centroid to the three vertices be d1 = 4 cm, d2 = 6 cm, and d3 = 8 cm.For each distance d, the full median length m = (3 / 2) * d.So corresponding medians are m1 = (3 / 2) * 4 = 6 cm.m2 = (3 / 2) * 6 = 9 cm.m3 = (3 / 2) * 8 = 12 cm.The smallest median among 6, 9, and 12 is 6 cm.

Verification / Alternative check:
To verify the ratio, recall the centroid property. The centroid is the balance point of the triangle, and medians are line segments joining vertices to mid points of opposite sides. Each median is split in the ratio 2 : 1 starting from the vertex. If a median is 6 cm long, then the segment from the vertex to the centroid is (2 / 3) * 6 = 4 cm, and the remaining part is 2 cm, matching this ratio. Similarly, medians of 9 cm and 12 cm give vertex to centroid distances of 6 cm and 8 cm respectively. Thus the conversion formula m = (3 / 2) * d is consistent and valid.

Why Other Options Are Wrong:
Option 8 cm corresponds to confusing the distance from the centroid to a vertex with the full median length for d = 4 or mixing up which value is smallest. Option 7 cm is not equal to (3 / 2) of any of the given distances. Option 5 cm similarly does not match any median derived from the 4, 6, 8 distances. Option 9 cm is one of the medians but not the smallest one; the question specifically asks for the smallest median.

Common Pitfalls:
Students sometimes reverse the ratio and assume that the centroid divides the median in the ratio 1 : 2 instead of 2 : 1 from the vertex. Others may treat the given distances as the entire medians rather than just the vertex to centroid segments. This leads to answers 4, 6, or 8 instead of their scaled versions. Remembering that the centroid is twice as close to each vertex as it is to the mid point of the opposite side helps keep the ratio 2 : 1 clear.

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