Geometry

Question Analysis

This question focuses on applying the properties of the centroid of a triangle. The key is to use the  ratio in which the centroid divides the median to set up an equation involving , and then find the length of .

 

Key Concept Explanation

The centroid of a triangle divides each median in the ratio . In , since  is the centroid and  is a median,  (the segment from the vertex to the centroid  is twice the segment from the centroid to the mid - point ), which also means .

 

Step - by - Step Solution

1. Set up the equation based on the centroid - median ratio:

Given  and , and using the ratio , we substitute the expressions:

.

2. Solve the equation for :

Expand the right - hand side: .

Move the  terms to one side and constants to the other: .

Combine like terms: