In a `triangle ABC`. AB = 5 cm, AC = 7 cm. If AD is the angle bisector of `angle A, then BD :CD is :

Options

A. 25:49
B. 49:25
C. 6:1
D. 5:7

Answer

To find the ratio \( BD : CD \) in triangle \( ABC \) where \( AB = 5 \, \text{cm} \), \( AC = 7 \, \text{cm} \), and \( AD \) is the angle bisector of \( \angle A \), we can use the Angle Bisector Theorem.

### Step-by-Step Solution:

1. **Understand 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. In this case, it can be expressed as:
\[ \frac{BD}{CD} = \frac{AB}{AC} \]

2. **Substitute the Known Values**:
From the problem, we know:
- \( AB = 5 \, \text{cm} \)
- \( AC = 7 \, \text{cm} \)

Now, substituting these values into the ratio:
\[ \frac{BD}{CD} = \frac{5}{7} \]

3. **Conclusion**:
Therefore, the ratio \( BD : CD \) is:
\[ BD : CD = 5 : 7 \]