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, , which also means and .

 

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: