The line segment joining the midpoints of the diagonals of a trapezium are parallel to each of the parallel sides and is equal to ______ the difference of these sides,

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(1)/(4)`
D. `(3)/(4)`

Answer

To solve the problem, we need to analyze the trapezium and the properties of the midpoints of its diagonals. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Draw the Trapezium**:
- Let’s denote the trapezium as ABCD, where AB is parallel to CD.

2. **Identify the Diagonals**:
- The diagonals of the trapezium are AC and BD.

3. **Find the Midpoints**:
- Let E be the midpoint of diagonal AC and F be the midpoint of diagonal BD.

4. **Connect the Midpoints**:
- Draw the line segment EF connecting points E and F.

5. **Establish Parallelism**:
- By the properties of trapeziums, the line segment EF is parallel to the bases AB and CD.

6. **Use the Midpoint Theorem**:
- According to the Midpoint Theorem, the length of the segment EF is equal to half the difference of the lengths of the parallel sides (AB and CD).

7. **Write the Relationship**:
- Therefore, we can express this relationship mathematically as:
\[ EF = \frac{1}{2} |AB - CD| \]

8. **Conclusion**:
- The line segment joining the midpoints of the diagonals of a trapezium is parallel to each of the parallel sides and is equal to **half** the difference of these sides.

### Final Answer:
The answer is **1/2** the difference of these sides.

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