In triangle ABC, AD is a median and G is the centroid of the triangle lying on AD. What is the ratio AG : AD?

Introduction / Context:
This is a pure concept question in triangle geometry about the centroid and medians. The centroid is one of the most important centres of a triangle, and it has a specific ratio property along medians. The question asks for the ratio in which the centroid divides a median drawn from a vertex to the midpoint of the opposite side. Understanding this property is essential for many geometric proofs and coordinate geometry applications.

Given Data / Assumptions:

- ABC is any triangle.
- AD is a median, so D is the midpoint of side BC and AD connects vertex A to D.
- G is the centroid of triangle ABC and lies on the median AD.
- By definition, the centroid is the intersection point of the three medians of the triangle.
- We are asked to find the ratio AG : AD, that is, how much of the median lies between A and G compared to the whole median from A to D.

Concept / Approach:
A well known property of the centroid is that it divides each median in the ratio 2 : 1, counting from the vertex to the midpoint of the opposite side. That is, if we label the median from A to the midpoint D, then AG : GD = 2 : 1. Since AD is the entire median and is composed of AG plus GD, we can express AD as three equal parts, with AG being two of those parts. This immediately gives the ratio AG : AD.

Step-by-Step Solution:
Step 1: Recall the definition of a median: AD is a line segment from vertex A to the midpoint D of side BC.Step 2: Recall the definition of the centroid G: the common intersection point of the three medians of a triangle.Step 3: A key property of the centroid is that it divides each median in the ratio 2 : 1, with the longer part closer to the vertex.Step 4: On median AD, this means AG : GD = 2 : 1.Step 5: Let each unit in this ratio represent one part. Then AD = AG + GD = 2 parts + 1 part = 3 parts.Step 6: Therefore AG is 2 parts out of the total 3 parts of AD.Step 7: Hence AG : AD = 2 : 3.

Verification / Alternative check:
You can visualise this by placing triangle ABC in coordinate geometry. Suppose A, B and C have coordinates and D is the midpoint of BC. The centroid G is the average of the coordinates of A, B and C. If you compute vector AG and compare it to AD, you will find that AG is always two thirds of AD, regardless of the triangle shape. This coordinate demonstration confirms the ratio AG : AD = 2 : 3.

Why Other Options Are Wrong:
The ratio 3 : 2 reverses the correct ordering, implying that the centroid is closer to the midpoint than to the vertex, which is not true. Ratios 3 : 1 and 1 : 2 would suggest that the centroid either lies much closer to the vertex than two thirds of the way or exactly at the midpoint, both of which contradict the established centroid property. The ratio 1 : 3 also has no basis in standard geometry for centroids.

Common Pitfalls:
Learners often confuse properties of the centroid with those of other centres like the incenter or orthocentre. Some mistakenly think the centroid lies at the midpoint of each median, giving a 1 : 2 ratio rather than 2 : 1. It is important to remember that the centroid is located two thirds of the distance from each vertex to the midpoint of the opposite side, not halfway along the median.

Final Answer:
The required ratio is AG : AD = 2 : 3.