Let BE and CF be the two medians of a` triangle`ABC and G be the intersection. Also, let EF cut AG at O, then AO : OG is

Options

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

Answer

To solve the problem, we need to determine the ratio \( AO : OG \) where \( G \) is the centroid of triangle \( ABC \) formed by the medians \( BE \) and \( CF \), and \( O \) is the intersection of line \( EF \) with line \( AG \).

### Step-by-Step Solution:

1. **Understanding the Medians**:
- In triangle \( ABC \), \( BE \) and \( CF \) are medians. A median connects a vertex to the midpoint of the opposite side. Therefore, \( E \) is the midpoint of \( AC \) and \( F \) is the midpoint of \( AB \).

2. **Finding the Centroid**:
- The centroid \( G \) of triangle \( ABC \) is the point where the three medians intersect. It divides each median in the ratio \( 2:1 \). This means that if we take median \( AG \), then \( AG \) is divided by \( G \) such that \( AG : GB = 2 : 1 \).

3. **Identifying Points**:
- Let \( O \) be the point where line \( EF \) intersects line \( AG \). We need to find the ratio \( AO : OG \).

4. **Using the Properties of the Centroid**:
- Since \( G \) divides the median \( AG \) in the ratio \( 2:1 \), we can express this as:
\[ \frac{AO}{OG} = \frac{2}{1} \]
- This means that for every 2 parts of \( AO \), there is 1 part of \( OG \).

5. **Conclusion**:
- Therefore, the ratio \( AO : OG \) is \( 2 : 1 \).

### Final Answer:
\[ AO : OG = 2 : 1 \]