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Which of the hatched rectangles has the larger content?

A Rectangle 1

B Rectangle 2

C The volumes of the two rectangles are equal

D The rectangles have the same content if and only if I is the centre of SQ

E The rectangles have the same content if and only if PQRS is a square

To determine which of the hatched rectangles has the larger area (content), we need to analyze the diagram and the given options. Here's the analysis step by step:

### Step 1: Identify the Geometry
- The figure is a quadrilateral \(PQRS\) with a diagonal \(PR\).
- \(I\) is the intersection point of the diagonal \(PR\) and line \(SQ\).
- Rectangles 1 and 2 are formed on either side of \(PR\) within the quadrilateral.

### Step 2: Consider the Area of Rectangles
- **Rectangle 1** lies within \(PSQI\).
- **Rectangle 2** lies within \(IRSQ\).
- Both rectangles have bases that are segments of \(PS\) and \(SR\), respectively, and their heights extend up to \(PR\).

### Step 3: Analyze the Options
1. **Option A (Rectangle 1 has a larger area):** This would be true if \(PS > SR\), implying the area is larger.
   
2. **Option B (Rectangle 2 has a larger area):** This would be true if \(SR > PS\), implying the area is larger.
   
3. **Option C (Areas are equal):** This suggests the two rectangles have equal bases and heights, leading to equal areas. This would only occur if \(PQRS\) has certain symmetrical properties, but without specific information, we cannot assume this.
   
4. **Option D (Areas are equal if \(I\) is the center of \(SQ\)):** This suggests a particular geometric condition where the intersection of the diagonal at the midpoint would equalize the areas, requiring symmetry about the diagonal.
   
5. **Option E (Areas are equal if \(PQRS\) is a square):** In a square, the diagonal divides the square into two equal-area triangles, making the areas of the rectangles equal.

### Conclusion:
**Option E** is correct because if \(PQRS\) is a square, the diagonal \(PR\) would divide the square into two equal areas, making the areas of Rectangles 1 and 2 equal.

Would you like to go into more details or have any questions? 

Here are some follow-up questions:

1. What is the geometric significance of a diagonal in a square?
2. How can we prove that the areas of two rectangles are equal in a square?
3. What conditions ensure that a point \(I\) on a diagonal bisects two areas equally?
4. Can you determine the area of the rectangles without knowing the side lengths?
5. How does the position of point \(I\) affect the areas of the rectangles?
6. What role does symmetry play in determining the areas of shapes within a square?
7. Could the areas of rectangles still be equal if \(PQRS\) is a rhombus?
8. How would the analysis change if \(PQRS\) was not a square but a general rectangle?

**Tip:** When working with geometric figures, it's essential to consider any symmetry and special points like midpoints, which often simplify the analysis.

Explore how to determine which hatched rectangle has a larger area using geometric analysis and diagonal intersections. Understand the conditions where rectangles have equal areas and the geometric significance of diagonals in shapes. Suitable for Grades 10-12.

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