If AD is the median of the triangle ABC and G be the centroid then the ratio of AG : AD is.
यदि AD त्रिभुज ABC की माध्यिका है और G केंद्रक है, तो AG : AD का अनुपात क्या होगा?

Options

A. `1:3`
B. `2:1`
C. `3:2`
D. `2:3`

Explanation

To solve the problem, we need to find the ratio of AG to AD, where AD is the median of triangle ABC and G is the centroid.

### Step-by-Step Solution:

1. **Understanding the Triangle and Median**:
- Consider triangle ABC. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Here, AD is the median from vertex A to the midpoint D of side BC.

2. **Identifying the Centroid**:
- The centroid (G) of a triangle is the point where the three medians intersect. It divides each median into two segments: one segment is twice the length of the other. Specifically, the segment connecting the vertex to the centroid is twice as long as the segment connecting the centroid to the midpoint of the opposite side.

3. **Using the Property of the Centroid**:
- The centroid divides the median in the ratio 2:1. This means that if we denote AG as the segment from vertex A to the centroid G, and GD as the segment from the centroid G to the midpoint D, we have:
\[ AG : GD = 2 : 1 \]

4. **Finding the Ratio AG : AD**:
- The entire length of the median AD can be expressed as:
\[ AD = AG + GD \]
- Since we know AG : GD = 2 : 1, we can express GD in terms of AG:
\[ GD = \frac{1}{2} AG \]
- Therefore, we can express AD as:
\[ AD = AG + GD = AG + \frac{1}{2} AG = \frac{3}{2} AG \]

5. **Calculating the Ratio**:
- Now, we can find the ratio AG : AD:
\[ AG : AD = AG : \frac{3}{2} AG \]
- Simplifying this gives:
\[ AG : AD = 1 : \frac{3}{2} = 2 : 3 \]

### Final Answer:
The ratio of AG to AD is \( 2 : 3 \).