In the triangle ABC, the internal bisector of angle A meets the side BC at D. If BD=7.5 cm and BC=9 cm. Then calculate the value of AB:AC.

Options

A. 2 :1
B. 1 :2
C. 5 :1
D. 1 :3

Answer

To solve the problem, we will use the Angle Bisector Theorem, which 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.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Let \( BD = 7.5 \) cm
- Let \( BC = 9 \) cm

2. **Calculate \( DC \)**:
- Since \( BC \) is the total length of side \( BC \), we can find \( DC \) using the equation:
\[ DC = BC - BD \]
- Substituting the values:
\[ DC = 9 \, \text{cm} - 7.5 \, \text{cm} = 1.5 \, \text{cm} \]

3. **Apply the Angle Bisector Theorem**:
- According to the Angle Bisector Theorem:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]
- Substituting the values we found:
\[ \frac{AB}{AC} = \frac{7.5}{1.5} \]

4. **Simplify the Ratio**:
- To simplify \( \frac{7.5}{1.5} \):
\[ \frac{7.5 \div 1.5}{1.5 \div 1.5} = \frac{5}{1} \]

5. **Conclusion**:
- Therefore, the ratio \( AB:AC \) is:
\[ AB:AC = 5:1 \]

### Final Answer:
The value of \( AB:AC \) is \( 5:1 \).