In a `Delta ABC, AD ` is a median. The bisectors of `angle ADB and angle ADC` meets AB & AC at E and F respectively . AE:BE=3:4 . Find `EF:BC`.

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Triangle and Given Information:**
- We have triangle \( ABC \) with \( AD \) as the median. This means \( BD = CD \).
- The angle bisectors of \( \angle ADB \) and \( \angle ADC \) meet \( AB \) and \( AC \) at points \( E \) and \( F \) respectively.
- We are given that \( AE:BE = 3:4 \).

2. **Using the Angle Bisector Theorem:**
- According to the Angle Bisector Theorem, the angle bisector divides the opposite side in the ratio of the other two sides.
- For triangle \( ABD \), since \( DE \) is the angle bisector of \( \angle ADB \), we have:
\[ \frac{AD}{BD} = \frac{AE}{BE} \]
- Given \( AE:BE = 3:4 \), we can express this as:
\[ \frac{AE}{BE} = \frac{3}{4} \]
- Therefore, we can write:
\[ \frac{AD}{BD} = \frac{3}{4} \quad \text{(1)} \]

3. **Applying the Angle Bisector Theorem Again:**
- For triangle \( ACD \), since \( DF \) is the angle bisector of \( \angle ADC \), we have:
\[ \frac{AD}{CD} = \frac{AF}{FC} \]
- Since \( BD = CD \) (because \( AD \) is a median), we can replace \( CD \) with \( BD \):
\[ \frac{AD}{BD} = \frac{AF}{FC} \quad \text{(2)} \]

4. **Equating the Ratios:**
- From equations (1) and (2), we can equate the right-hand sides:
\[ \frac{AE}{BE} = \frac{AF}{FC} \]
- This means:
\[ \frac{3}{4} = \frac{AF}{FC} \]

5. **Using the Basic Proportionality Theorem:**
- Since \( \frac{AE}{BE} = \frac{AF}{FC} \), by the converse of the Basic Proportionality Theorem, we conclude that \( EF \) is parallel to \( BC \).
- This implies that triangles \( AEF \) and \( ABC \) are similar.

6. **Finding the Ratios of Corresponding Sides:**
- From the similarity of triangles \( AEF \) and \( ABC \), we have:
\[ \frac{AE}{AB} = \frac{EF}{BC} \]
- We know \( AE = 3 \) and \( BE = 4 \), thus:
\[ AB = AE + BE = 3 + 4 = 7 \]
- Therefore:
\[ \frac{AE}{AB} = \frac{3}{7} \]
- Substituting this into the similarity ratio gives:
\[ \frac{3}{7} = \frac{EF}{BC} \]

7. **Final Ratio:**
- Thus, we find:
\[ EF:BC = 3:7 \]

### Conclusion:
The ratio \( EF:BC \) is \( 3:7 \).