In a `DeltaABC, AD` is angle bisector , `AB=7 cm , AC=8 cm, BC=6cm`, If I is the incentre . Find AI/ID.

Answer

To solve the problem of finding the ratio \( \frac{AI}{ID} \) in triangle \( ABC \) where \( AD \) is the angle bisector, we can use the Angle Bisector Theorem.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \( AB = c = 7 \, \text{cm} \)
- Let \( AC = b = 8 \, \text{cm} \)
- Let \( BC = a = 6 \, \text{cm} \)

2. **Apply the Angle Bisector Theorem**:
The Angle Bisector Theorem states that the ratio of the lengths of the two segments created by the angle bisector on the opposite side is equal to the ratio of the lengths of the other two sides of the triangle. Therefore, we have:
\[ \frac{AI}{ID} = \frac{AB}{AC} = \frac{c}{b} \]

3. **Substitute the values**:
Substitute the lengths of \( AB \) and \( AC \) into the equation:
\[ \frac{AI}{ID} = \frac{7}{8} \]

4. **Final Ratio**:
Thus, the ratio \( \frac{AI}{ID} \) is:
\[ \frac{AI}{ID} = \frac{7}{8} \]

### Conclusion:
The ratio \( \frac{AI}{ID} \) is \( 7:8 \).