In a triangle ABC, AB = 5 cm, BC = 6 cm and CA = 7 cm. If bisectors of `angleA, angleB, angleC` respectively meet sides BC, CA and AB at D. E, F and I be the incentre of the triangle then
What is BD : DC ?

Options

A. `5:7`
B. `7:5`
C. `5:6`
D. `6:5`

Answer

To find the ratio \( BD : DC \) in triangle \( ABC \) where \( AB = 5 \, \text{cm} \), \( BC = 6 \, \text{cm} \), and \( CA = 7 \, \text{cm} \), we will use the Angle Bisector Theorem.

### Step-by-Step Solution:

1. **Identify the Sides of the Triangle**:
- Let \( AB = c = 5 \, \text{cm} \)
- Let \( BC = a = 6 \, \text{cm} \)
- Let \( CA = b = 7 \, \text{cm} \)

2. **Apply the Angle Bisector Theorem**:
- The Angle Bisector Theorem states that the angle bisector divides the opposite side in the ratio of the adjacent sides.
- For angle \( A \) bisector meeting \( BC \) at point \( D \), we have:
\[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{c}{b} \]

3. **Substitute the Values**:
- Substitute the lengths of sides \( AB \) and \( AC \):
\[ \frac{BD}{DC} = \frac{5}{7} \]

4. **Conclusion**:
- Therefore, the ratio \( BD : DC \) is \( 5 : 7 \).

### Final Answer:
\[ BD : DC = 5 : 7 \]