Suppose that the medians BD, CE and AF of triangle ABC meet at the centroid G. What is the ratio AG : GF?

Introduction / Context:
This conceptual geometry question is about medians and the centroid of a triangle. It tests a standard property known in coordinate geometry and Euclidean geometry: the centroid divides each median of a triangle in a fixed ratio. Understanding this property is very useful for solving many triangle related problems in competitive examinations.

Given Data / Assumptions:
• ABC is any triangle.• BD, CE, and AF are medians of triangle ABC.• All three medians meet at point G, which is called the centroid.• We need the ratio AG : GF along the median from vertex A to side BC.

Concept / Approach:
A median of a triangle joins a vertex to the midpoint of the opposite side. In any triangle, the three medians are concurrent; they meet at a single point called the centroid. A fundamental result in geometry states that the centroid divides each median in the ratio 2:1, with the longer segment always between the vertex and the centroid. Therefore, on median AF from vertex A to midpoint F of BC, the centroid G lies such that AG:GF = 2:1.

Step-by-Step Solution:
Step 1: Recall the definition of a median.A median connects a vertex to the midpoint of the opposite side.Step 2: Recall the property of the centroid.The centroid is the common point of intersection of the three medians.Step 3: Use the known ratio property.For any median, the centroid divides the median such that the vertex to centroid segment is twice the centroid to midpoint segment.Step 4: Apply this to median AF.Hence, AG:GF = 2:1.

Verification / Alternative check:
In coordinate geometry, if the vertices of a triangle are (x1, y1), (x2, y2), and (x3, y3), then the centroid has coordinates ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3). If we compute the midpoint of BC and use the formula for the centroid, we see that the vector from A to G is two thirds of the vector from A to F, and the vector from G to F is one third. This gives the same ratio of 2:1, confirming the property.

Why Other Options Are Wrong:
Option A: 1:2 reverses the ratio and would imply the centroid is closer to the midpoint, which contradicts the known theorem.Option C: 1:3 does not correspond to any standard median division property.Option D: 2:3 is not supported by geometric theory for medians and centroids.

Common Pitfalls:
Some learners misremember the ratio as 1:2 or confuse it with other properties such as side ratios in special triangles. Others may incorrectly think the centroid divides the median into equal halves. To avoid this, it is helpful to remember that the centroid is always two thirds of the way from a vertex to the midpoint of the opposite side, giving a fixed 2:1 division along every median.

Final Answer:
The ratio AG : GF is 2:1.