Explanation:

To understand why the centroid of a triangle divides the median in the ratio 2:1, let's first define what a centroid and a median are.

Centroid:
The centroid of a triangle is the point of intersection of its three medians. A median is a line segment joining a vertex of a triangle to the midpoint of the opposite side.

Median:
In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposing side. Each triangle has three medians, one drawn from each vertex.

The Centroid and Medians:
1. The centroid is always located inside the triangle.
2. The centroid divides each median into two segments.
3. The ratio of the lengths of these two segments is always 2:1.

Proof:
To prove that the centroid divides the median in a 2:1 ratio, we can use the concept of vectors in geometry.

Consider a triangle ABC with vertices A, B, and C. Let D be the midpoint of side BC. The centroid of the triangle is denoted by point G.
Let's use vectors to denote the position of each point. Let vector A represent the position of point A, vector B represent the position of point B, and so on.

Now, we know that the centroid is the point of intersection of its medians. The median from vertex A passes through the midpoint D of side BC.

The position vector of the centroid G can be calculated as:
G = (A + B + C) / 3

Similarly, the position vector of the midpoint D can be calculated as:
D = (B + C) / 2

Now, let's find the vector GD:
GD = G - D = (A + B + C) / 3 - (B + C) / 2

Simplifying the above equation, we get:
GD = (2A - B - C) / 6

This means that the vector GD is one-sixth of the vector from A to B and C.

Since the vector GD is one-sixth of the vector from A to B and C, we can conclude that the length of GD is one-sixth of the length of AD.

Therefore, the centroid G divides the median AD in the ratio 2:1.

Hence, the correct answer is option D - 2:1.