In `Delta ABC,D` is the midpoint of AC, F is a point AB such that CF bisects BD at E. If AF = 18 cm, then the length of BF is :

Options

A. 12 cm
B. 8 cm
C. 6 cm
D. 9 cm

Answer

To solve the problem step by step, we will analyze the given information and apply geometric principles.

### Step-by-Step Solution:

1. **Understanding the Triangle and Points**:
- We have triangle ABC.
- D is the midpoint of AC, which means AD = DC.
- F is a point on line segment AB.
- CF bisects BD at point E.

**Hint**: Visualize the triangle and label the points to understand their relationships.

2. **Given Information**:
- We are given that AF = 18 cm.
- We need to find the length of BF.

**Hint**: Write down all the known lengths and relationships in the triangle.

3. **Using the Midpoint Theorem**:
- Since D is the midpoint of AC, we can use properties of midpoints and bisectors.
- CF bisects BD at E, which implies that BE = ED.

**Hint**: Remember that a bisector divides a segment into two equal parts.

4. **Setting Up Relationships**:
- Let BF = x cm.
- Since AF + BF = AB, we have:
\[ AF + BF = 18 + x \]
- Since D is the midpoint of AC, we can say that the segments created by the bisector will maintain proportionality.

**Hint**: Use the properties of triangles and segments to set up equations.

5. **Using Parallelogram Properties**:
- Since we have established that BF is parallel to PD and FA is also parallel to PD, we can conclude that BFDP forms a parallelogram.
- In a parallelogram, opposite sides are equal. Therefore, we have:
\[ PD = AF/2 = 18/2 = 9 \text{ cm} \]

**Hint**: Recall that in a parallelogram, opposite sides are equal in length.

6. **Finding BF**:
- Since PD = BF, we can conclude:
\[ BF = PD = 9 \text{ cm} \]

**Hint**: Substitute the known values to find the unknown length.

7. **Final Answer**:
- Therefore, the length of BF is 9 cm.

### Conclusion:
The length of BF is **9 cm**.