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The problem involves the triangle \(ABC\) with medians \(AD\) and \(BE\). The intersection of the medians is point \(G\), and \(F\) is a point on \(AG\) such that \(FG = \frac{1}{3} AG\). We are asked to find the value of the ratio \(\frac{AF}{AD}\).

### Solution:
1. In any triangle, the medians intersect at the centroid \(G\), which divides each median into two segments in the ratio \(2:1\), with the longer segment closer to the vertex.
2. Here, \(F\) is on \(AG\), and it is given that \(FG = \frac{1}{3} AG\). 
   
   Since \(FG\) is \(\frac{1}{3}\) of \(AG\), the other segment \(AF\) must be \(\frac{2}{3}\) of \(AG\).
3. The length of the entire median \(AD\) is \(AG + GD\), and since \(G\) divides \(AD\) in the ratio \(2:1\), we have:
   \[
   AG = \frac{2}{3} AD \quad \text{and} \quad GD = \frac{1}{3} AD.
   \]
4. Since \(AF = \frac{2}{3} AG\), substituting \(AG\) in terms of \(AD\), we get:
   \[
   AF = \frac{2}{3} \times \frac{2}{3} AD = \frac{4}{9} AD.
   \]
5. Therefore, the ratio \(\frac{AF}{AD}\) is:
   \[
   \frac{AF}{AD} = \frac{4}{9}.
   \]

Thus, the correct answer is:

**c) \(\frac{4}{9}\)**.

Would you like more details or have any questions?

Here are some related questions:
1. How do you calculate the centroid of a triangle?
2. What are the properties of medians in a triangle?
3. How does the centroid divide each median of a triangle?
4. How can we use ratios to solve geometric problems involving medians?
5. What other important points of concurrency exist in a triangle?

**Tip:** In any triangle, the centroid divides each median in a consistent \(2:1\) ratio, which is a key property for solving many problems involving medians.

In the triangle ABC with medians AD, and BE, point G is the intersection of the medians, and F is a point on AG such that FG = 1/3 AG. What is the value of the ratio AF/AD?

Explore the solution to calculate the ratio AF/AD in triangle ABC using centroid and median properties. This guide explains the application of the centroid theorem and ratios in solving geometry problems, suitable for high school students in grades 9-12.

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