Thank you for the clarification that the diagonal is \( SQ \) instead of \( PR \). With this new information, let's reassess the situation.

### Step 1: Identify the Geometry with the Correct Diagonal
- The quadrilateral \( PQRS \) has a diagonal \( SQ \).
- Point \( I \) is where \( SQ \) intersects \( PR \).
- Rectangles 1 and 2 are situated on either side of the diagonal \( SQ \).

### Step 2: Consider the Areas of Rectangles Based on \( SQ \)
- **Rectangle 1** lies on the left side of \( SQ \), within \( PSQI \).
- **Rectangle 2** lies on the right side of \( SQ \), within \( IRQS \).

### Step 3: Analyze the Options with the Correct Diagonal
1. **Option A (Rectangle 1 has a larger area):** This could be true if the base \( PI \) and height \( PS \) are larger.
   
2. **Option B (Rectangle 2 has a larger area):** This could be true if the base \( IQ \) and height \( RS \) are larger.
   
3. **Option C (Areas are equal):** This suggests symmetry along the diagonal \( SQ \), leading to equal areas.
   
4. **Option D (Areas are equal if \( I \) is the center of \( SQ \)):** This implies that if \( I \) is the midpoint of \( SQ \), the rectangles could have equal areas, as both would be symmetrically positioned.

5. **Option E (Areas are equal if \( PQRS \) is a square):** This would mean that if \( PQRS \) is a square, the diagonal \( SQ \) would divide it into two equal-area triangles, making the areas of the rectangles equal.

### Conclusion:
With the diagonal being \( SQ \), the **correct answer is Option D**. The rectangles have the same area if and only if \( I \) is the center (midpoint) of the diagonal \( SQ \). This is because, at the midpoint, the division creates two equal areas on either side of the diagonal.

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How does the position of point \( I \) affect the areas of Rectangles 1 and 2?
2. What is the significance of the midpoint on a diagonal in a quadrilateral?
3. Can the areas of two rectangles still be equal if \( I \) is not the center of \( SQ \)?
4. How can you prove that the areas of Rectangles 1 and 2 are equal when \( I \) is the midpoint of \( SQ \)?
5. What happens to the areas if \( PQRS \) is a rectangle instead of a square?
6. Why does symmetry along the diagonal lead to equal areas for Rectangles 1 and 2?
7. How would you calculate the areas of these rectangles if you knew the lengths of \( PS \), \( SQ \), and \( SR \)?
8. What other conditions might make the areas of these two rectangles equal?

**Tip:** When working with diagonals in quadrilaterals, always consider the properties of midpoints and symmetry, as these often determine equal areas or other geometric relationships.

Explore the analysis of a quadrilateral problem where the diagonal SQ affects the areas of two rectangles. Understand how the position of point I along diagonal SQ impacts the areas of Rectangles 1 and 2. Discover the conditions under which these areas are equal or unequal. Suitable for students studying advanced geometry.

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