The figure below is made up of two identical squares, ABCD and GJKH, and 2 identical rectangles, EFDA and BCHG. The perimeter of EBCF is 56 cm and the perimeter of AJKD is 72 cm. What is the area of square ABCD? Answer: ___ $\text{cm}^2$

1. Define variables for the dimensions of the rectangles and squares

Let the side length of the identical squares $ABCD$ and $GJKH$ be $x$ cm.
Let the width of the identical rectangles $EFDA$ and $BCHG$ be $y$ cm.
Since the rectangles share the height with the squares, their length is $x$ cm.

2. Form equations based on the given perimeters

Perimeter of $EBCF$ (composed of rectangle $EFDA$ and square $ABCD$):
The length of $EBCF$ is $y + x$ and its width is $x$.
Perimeter = $2(y + x + x) = 2(y + 2x) = 2y + 4x = 56$.

Perimeter of $AJKD$ (composed of square $ABCD$, rectangle $BCHG$, and square $GJKH$):
The length of $AJKD$ is $x + y + x = 2x + y$ and its width is $x$.
Perimeter = $2(2x + y + x) = 2(3x + y) = 6x + 2y = 72$.

3. Solve the simultaneous equations to find the side length of the square

We have the system:

1. $4x + 2y = 56$
2. $6x + 2y = 72$

Subtracting equation (1) from equation (2):
$(6x + 2y) - (4x + 2y) = 72 - 56$
$2x = 16$
$x = 8$

4. Calculate the area of square ABCD

Area of square $ABCD = x \times x = 8 \times 8 = 64$ cm$^2$.