`A B C D` is a parallelogram and `E` is the mid-point of `B C ,\ D E\ a n d\ A B` when produced meet at `F`. Then `A F=`

Options

A. `3/2A B`
B. `2 A B`
C. `3 A B`
D. `5/4 A B`

Answer

To solve the problem, we need to find the length of \( AF \) in the given parallelogram \( ABCD \) where \( E \) is the midpoint of \( BC \) and lines \( DE \) and \( AB \) when produced meet at point \( F \).

### Step-by-Step Solution:

1. **Identify the Properties of the Parallelogram:**
- In parallelogram \( ABCD \), opposite sides are equal, i.e., \( AB = CD \) and \( AD = BC \).
- Angles opposite to each other are equal, i.e., \( \angle ABC = \angle ADC \) and \( \angle BCD = \angle DAB \).

2. **Locate the Midpoint:**
- Since \( E \) is the midpoint of \( BC \), we can say that \( BE = EC \).

3. **Analyze the Triangles:**
- Consider triangles \( DCE \) and \( FBE \).
- We know that \( \angle DEC = \angle FEB \) (vertically opposite angles).
- Also, \( \angle DCB = \angle EBF \) (alternate interior angles, since \( DC \parallel AB \)).

4. **Use the Midpoint Property:**
- Since \( E \) is the midpoint, we have \( CE = BE \).

5. **Apply AAS Congruence:**
- From the above information, we can conclude that triangles \( DCE \) and \( FBE \) are congruent by the AAS (Angle-Angle-Side) congruence criterion:
- \( \angle DEC = \angle FEB \)
- \( \angle DCB = \angle EBF \)
- \( CE = BE \)

6. **Corresponding Parts of Congruent Triangles:**
- Since \( \triangle DCE \cong \triangle FBE \), we have:
- \( DC = FB \) (corresponding sides)

7. **Relate \( AB \) and \( FB \):**
- Since \( AB = CD \) (opposite sides of the parallelogram), we can say \( AB = FB \).

8. **Express \( AF \):**
- We know that \( AF = AB + BF \).
- Since \( BF = AB \), we can substitute:
- \( AF = AB + AB = 2AB \).

9. **Conclusion:**
- Thus, we find that \( AF = 2AB \).

### Final Answer:
\[ AF = 2AB \]