In a `DeltaABC`, BC = 9 cm, AC = 40 cm and AB = 41 cm. If bisector of angle A meets side BC at D then ratio of area of `DeltaABD` and `DeltaABC` is

Options

A. `41:40`
B. `9:40`
C. `9:41`
D. `41:81`

Answer

To solve the problem, we need to find the ratio of the areas of triangles \( \Delta ABD \) and \( \Delta ABC \).

### Step-by-Step Solution:

1. **Identify the Given Values:**
- \( BC = 9 \, \text{cm} \)
- \( AC = 40 \, \text{cm} \)
- \( AB = 41 \, \text{cm} \)

2. **Use the Angle Bisector Theorem:**
According to the Angle Bisector Theorem, the ratio of the segments created by the angle bisector on the opposite side is equal to the ratio of the other two sides. Here, \( D \) is the point where the angle bisector of \( \angle A \) meets side \( BC \).
\[ \frac{BD}{DC} = \frac{AB}{AC} \]
Substituting the known values:
\[ \frac{BD}{DC} = \frac{41}{40} \]

3. **Let \( BD = x \) and \( DC = 9 - x \):**
From the ratio:
\[ \frac{x}{9 - x} = \frac{41}{40} \]

4. **Cross-Multiply to Solve for \( x \):**
\[ 40x = 41(9 - x) \]
Expanding the right side:
\[ 40x = 369 - 41x \]
Adding \( 41x \) to both sides:
\[ 40x + 41x = 369 \]
\[ 81x = 369 \]
Dividing both sides by 81:
\[ x = \frac{369}{81} = \frac{41}{9} \]

5. **Find \( DC \):**
Now, substituting back to find \( DC \):
\[ DC = 9 - x = 9 - \frac{41}{9} = \frac{81 - 41}{9} = \frac{40}{9} \]

6. **Calculate the Ratio of Areas:**
The area of a triangle can be expressed as:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Since both triangles \( \Delta ABD \) and \( \Delta ABC \) share the same height from point \( A \) to line \( BC \), we can express the ratio of their areas as:
\[ \frac{\text{Area of } \Delta ABD}{\text{Area of } \Delta ABC} = \frac{BD}{BC} = \frac{BD}{BD + DC} \]
Substituting the values we found:
\[ \frac{\text{Area of } \Delta ABD}{\text{Area of } \Delta ABC} = \frac{\frac{41}{9}}{9} = \frac{41}{81} \]

7. **Final Ratio:**
Therefore, the ratio of the area of triangle \( \Delta ABD \) to the area of triangle \( \Delta ABC \) is:
\[ \frac{41}{81} \]

### Conclusion:
The ratio of the area of \( \Delta ABD \) to the area of \( \Delta ABC \) is \( 41:81 \).