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.