Dona Lígia has a square plot of land. She decides to divide it into five regions, consisting of four rectangles and one square as illustrated in the figure below:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-20.jpg?height=416&width=417&top_left_y=503&top_left_x=725)

$\mathrm{In}$ the figure above, we have that:

- The central square has an area of $64 \mathrm{~m}^{2}$;
- The longer sides of the four rectangles have the same length;
- The five regions have the same perimeter.

Determine the area of Dona Lígia's plot of land.
Solution

Since the area of the central square is equal to $64 \mathrm{~m}^{2}$, its side length is $8 \mathrm{~m}$. Since the perimeter of a square is four times the length of its side, we conclude that the perimeter of the central square is equal to $32 \mathrm{~m}$.

Since the five regions have the same perimeter, we conclude that the perimeter of each rectangle is also equal to $32 \mathrm{~m}$. Now observe the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_d18780f0fd52e0502cf8g-49.jpg?height=466&width=525&top_left_y=1857&top_left_x=868)

From this, we see that $M A + A N$ is equal to half the perimeter of rectangle MANS. Therefore,

$$
M A + A N = 16 \mathrm{~m}
$$

But the longer sides of the rectangles are equal, i.e., $M A = N B$. Thus, we can substitute $M A$ with $N B$ in equation (.4) to get $N B + A N = 16 \mathrm{~m}$. We conclude then that the side of the plot measures $N B + A N = 16 \mathrm{~m}$. Since the plot is square-shaped, the area of the plot is $(16 \mathrm{~m})^{2} = 256 \mathrm{~m}^{2}$.