In the given problem, we are told that $G$ is the centroid of $\triangle ABC$, and we are asked to find the length of $GE$.

Key Information:

• $G$ is the centroid, which means it divides each median into a 2:1 ratio, with the longer segment being closer to the vertex.
• $BG = 6$, which means $BG$ is two-thirds of the length of the median $BE$.
• $AF = 12$, and $AE = 15$.

Let's solve this step by step:

1. Centroid Properties:

   • Since $G$ is the centroid, it divides the median $BE$ into two parts: $BG$ (the part from $B$ to $G$) and $GE$ (the part from $G$ to $E$).
   • The ratio of the centroid division is 2:1. Thus, $BG = 2 \times GE$.

2. Finding $GE$: Since $BG = 6$, we can use the 2:1 ratio to calculate $GE$. $BG = 2 \times GE \implies 6 = 2 \times GE$ Solving for $GE$: $GE = \frac{6}{2} = 3$

Thus, the length of $GE$ is 3 units.

Would you like further details or clarifications on any step?

Related Questions:

1. How do we find the coordinates of a centroid given the vertices of a triangle?
2. What is the significance of the 2:1 ratio in a centroid's division of medians?
3. How can we verify if a given point is the centroid of a triangle?
4. What are other properties of centroids in triangles?
5. How is the area of a triangle related to its centroid?

Tip:

The centroid always divides each median into a 2:1 ratio, and it is also the center of mass of a triangle!