A line EF is drawn parallel to the parallel sides AB and CD of a trapezium ABCD where E lies on AD and F lies on BC. If `AE:ED=2:1` then what is BF : BC ?

Options

A. A)`2:3`
B. B)`3:2`
C. C)`2:1`
D. D)`1:2`

Answer

To solve the problem, we will use the properties of similar triangles and the concept of ratios in trapeziums.

### Step-by-Step Solution:

1. **Understanding the Trapezium**:
- We have a trapezium ABCD with AB || CD (AB is parallel to CD).
- A line EF is drawn parallel to AB and CD, where E lies on AD and F lies on BC.

2. **Given Ratio**:
- We are given that AE:ED = 2:1. This means if we let AE = 2x, then ED = x.

3. **Using Thales' Theorem**:
- Since EF is parallel to AB and CD, by Thales' theorem, the segments on the sides AD and BC will be in the same ratio as AE and ED.
- Therefore, we can say that BF:FC = AE:ED = 2:1.

4. **Finding BF and BC**:
- Let BF = 2y and FC = y (since BF:FC = 2:1).
- Thus, BC = BF + FC = 2y + y = 3y.

5. **Finding the Ratio BF:BC**:
- Now we can find the ratio BF:BC.
- BF:BC = 2y:3y = 2:3.

6. **Final Answer**:
- Therefore, the ratio BF:BC is 2:3.