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The image you've provided contains a geometry problem involving a trapezium \(PQRS\) and some points on the trapezium.

Here’s the problem broken down:

### Problem Description:
- **Given**: \(PQRS\) is a trapezium.
  - \(PQ \parallel SR\)
  - \(M\) is the midpoint of \(PQ\)
  - \(N\) is the midpoint of \(SR\)
  - \(PQ = 12\), \(SR = 6\)
- **To Find**: 
  1. Prove that \(MN \parallel QR\).
  2. Prove that \(MN = \frac{1}{2} QR\).
  3. Calculate the area of the trapezium \(PQRS\).

### Solution:
1. **Prove \(MN \parallel QR\)**:
   - Since \(M\) and \(N\) are the midpoints of \(PQ\) and \(SR\) respectively, by the Midpoint Theorem, the line segment joining the midpoints of two sides of a trapezium is parallel to the third side. 
   - Therefore, \(MN \parallel QR\).

2. **Prove \(MN = \frac{1}{2} QR\)**:
   - By the Midpoint Theorem again, the length of the line segment joining the midpoints of two sides of a trapezium (i.e., \(MN\)) is half the length of the third side (i.e., \(QR\)).
   - Hence, \(MN = \frac{1}{2} QR\).

3. **Calculate the Area of Trapezium \(PQRS\)**:
   - The formula for the area of a trapezium is given by:
     \[
     \text{Area} = \frac{1}{2} \times (\text{Sum of parallel sides}) \times \text{Height}
     \]
   - In this case, the parallel sides are \(PQ = 12\) and \(SR = 6\).
   - To find the height, we use the fact that \(MN = \frac{1}{2} QR\). This implies that the height \(h\) can be calculated if we had \(QR\) or another side's information, but the height isn’t directly given, and \(MN\) is a mid-segment.

     If we assume that the height is perpendicular to the bases, we typically use coordinate geometry or more trapezoid properties for the exact height. With more context or information, we could find the exact height.

     - As a rough estimate:
       \[
       \text{Area} \approx \frac{1}{2} \times (12 + 6) \times h
       \]
     - But without the exact height, the final area cannot be determined exactly from the given data alone.

Would you like to proceed with a specific part of the question in more detail or require further explanation?

### Related Questions:
1. How can you determine the exact height of a trapezium if the lengths of all sides are known?
2. What is the significance of the Midpoint Theorem in trapezoids?
3. How do the properties of a trapezium differ from those of a parallelogram?
4. What methods can be used to find the area of a trapezium when the height is unknown?
5. Can the properties of the trapezium be applied to find other related line segments within it?

### Tip:
Always remember to check the properties of the geometric figure involved; they often provide shortcuts to solving the problem quickly and accurately.

Given trapezium PQRS with PQ parallel to SR, M and N are midpoints of PQ and SR, respectively. Prove that MN is parallel to QR, that MN equals half of QR, and find the area of the trapezium.

Learn how to prove that the mid-segment of trapezium PQRS is parallel to the non-parallel sides and how to calculate its area. This problem involves the Midpoint Theorem and is suitable for students in Grades 9-11.

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