In a parallelogram ABCD, AB= 20 cm and AD= 12cm. The bisector of angle A meets DC at E and BC produced at F. Find the length of CF.

Answer

To solve the problem step by step, we will use the properties of a parallelogram and the angle bisector theorem.

### Step-by-Step Solution:

1. **Draw the Parallelogram**:
- Start by drawing parallelogram ABCD where AB = 20 cm and AD = 12 cm.
- Since opposite sides of a parallelogram are equal, we have BC = 20 cm and DC = 12 cm.

2. **Identify the Points**:
- Mark the points A, B, C, and D accordingly.
- Let the angle bisector of angle A meet side DC at point E and the extended line of BC at point F.

3. **Assume the Length CF**:
- Let CF = x cm.
- Therefore, BC (which is equal to AD) can be expressed as BC = 12 cm + CF = 12 cm + x.

4. **Use the Angle Bisector Theorem**:
- According to the angle bisector theorem, the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the other two sides of the triangle.
- In triangle ABF, the angle bisector divides BF into segments BE and EF such that:
\[ \frac{AB}{AF} = \frac{BE}{EF} \]
- Here, AB = 20 cm and AF = BC = 12 cm + x.

5. **Set Up the Equation**:
- Since BE is equal to BC (12 cm) and EF is equal to CF (x), we can write:
\[ \frac{20}{12 + x} = \frac{12}{x} \]

6. **Cross-Multiply**:
- Cross-multiplying gives:
\[ 20x = 12(12 + x) \]

7. **Expand and Simplify**:
- Expanding the right side:
\[ 20x = 144 + 12x \]
- Rearranging gives:
\[ 20x - 12x = 144 \]
\[ 8x = 144 \]

8. **Solve for x**:
- Dividing both sides by 8:
\[ x = \frac{144}{8} = 18 \]

9. **Conclusion**:
- Therefore, the length of CF is 18 cm.

### Final Answer:
CF = 18 cm