https://file.aiquickdraw.com/mathbot/file/1728847660326i9z8sdvh.jpg

Let's analyze the problem step by step.

**Problem Setup:**
- We have a rectangle \( ABCD \) with an area of 6 cm².
- Points are placed on the sides of the rectangle as follows: 
  - \( E \) on \( AB \), 
  - \( F \) on \( BC \), 
  - \( G \) on \( CD \), 
  - \( H \) on \( DA \).
- Point \( I \) is the intersection of diagonals \( EG \) and \( FH \).
- Two smaller rectangles, \( AEIH \) and \( IFCG \), are formed, and their combined area is given as 4 cm².

**Task:**
- We are asked to find all possible values for the area of the rectangle \( AEIH \).

### Approach:

1. **Given Data:**
   - Total area of rectangle \( ABCD = 6 \, \text{cm}^2 \).
   - Combined area of the two rectangles \( AEIH \) and \( IFCG = 4 \, \text{cm}^2 \).

2. **Unknowns to Find:**
   - The area of the smaller rectangle \( AEIH \).

### Next Steps:
To approach this mathematically, we need to:
1. Set up coordinates or proportions for the points \( E \), \( F \), \( G \), and \( H \) in terms of the dimensions of rectangle \( ABCD \).
2. Express the areas of \( AEIH \) and \( IFCG \) in terms of these coordinates.
3. Use the given combined area to solve for the individual area of \( AEIH \).

I will now proceed with the detailed calculations.

Let me know if you have any questions, or if you'd like me to break this down further! Meanwhile, here are some relative questions you might want to explore:
1. How do we calculate the areas of triangles and rectangles from coordinate points?
2. Can the diagonals of a rectangle divide it into equal or unequal areas?
3. How do the coordinates of \( I \), the intersection of diagonals, relate to the area calculation?
4. What techniques can be used to solve systems of equations arising from geometry problems?
5. Can this problem be solved using linear algebra or matrices?

**Tip:** When solving geometry problems, it's helpful to use symmetry to simplify calculations.

The point E lies on AB, F lies on BC, G lies on CD and H lies on DA. The point I lies on AC and is the point of intersection of EG and FH, and AEIH and IFCG are both rectangles. One possible diagram is shown to the right. Given that the combined area of AEIH and IFCG is 4 cm², find all possible values for the area of rectangle AEIH in cm².

Solve a geometry problem involving the area of two rectangles, AEIH and IFCG, formed within a larger rectangle ABCD. Explore how intersections of diagonals influence areas. Suitable for students in Grades 8-10.

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