The problem involves a square paper with an area of 64 cm², from which four identical right-angled triangles were cut out. These triangles were rearranged to form a shape, and the perimeter of the shaded rectangle in the center is given as 20 cm. We are tasked with finding the area of the shaded rectangle.

Step 1: Dimensions of the Square

The area of the square is given as 64 cm². Since the area of a square is calculated as the side length squared, we can find the side length of the square: $\text{Side length of the square} = \sqrt{64} = 8 \text{ cm}$

Step 2: Properties of the Right-Angled Triangles

Since four identical right-angled triangles are cut from the square, we need to understand their dimensions. If we denote the legs of the triangle as $a$ and $b$, and the hypotenuse as $c$, the total area of the four triangles should be subtracted from the square's area to leave the shaded rectangle.

Step 3: Perimeter of the Shaded Rectangle

The perimeter of the shaded rectangle is given as 20 cm. If the dimensions of the rectangle are $l$ (length) and $w$ (width), the perimeter is: $2(l + w) = 20 \quad \Rightarrow \quad l + w = 10 \text{ cm}$

Step 4: Solving for the Area of the Shaded Rectangle

The area of the shaded rectangle is: $\text{Area of rectangle} = l \times w$ From the relationship $l + w = 10$, we still need one more piece of information, like either the length or width.

We also know that the triangles were cut from the square, which may help define $l$ and $w$ based on geometric properties. However, from the problem, no specific dimensions for the triangles or further geometrical constraints are explicitly stated.

Would you like me to further explore or infer the dimensions of the triangles and complete the calculation, or clarify the problem in another way?

Questions to expand:

1. How does the shape of the triangles affect the dimensions of the shaded rectangle?
2. Can we infer the length or width of the rectangle based on its perimeter?
3. What would happen if the triangles were isosceles right triangles?
4. How would changing the perimeter affect the area of the shaded rectangle?
5. Could the area of the shaded rectangle be found using another approach like coordinate geometry?

Tip: Always check whether a geometric figure can be divided into simpler parts to make calculations easier.