Geometry

Question Analysis

This question focuses on applying the properties of the centroid of a triangle. Given that  is the centroid and  is a median, we need to use the centroid - median ratio to find the length of .

 

Key Concept Explanation

The centroid of a triangle divides each median in the ratio . That is, the segment from the vertex to the centroid is twice as long as the segment from the centroid to the mid - point of the side. If  is a median and  is the centroid, then .

 

Step - by - Step Solution

1. Apply the centroid - median ratio:

We know that the ratio of the segment from the centroid to the mid - point of the side () to the entire median () is  (because the centroid divides the median in the ratio , so the part from the centroid to the mid - point is  of the median).

Given .

To find

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