A square is divided into 4 equal rectangles and one square, as shown in the figure below. The shaded square has an area of $36 \mathrm{~m}^{2}$. Each rectangle has an area of $216 \mathrm{~m}^{2}$. Determine the dimensions of the rectangles.

![](https://cdn.mathpix.com/cropped/2024_05_01_09dbad5281c800d666d6g-31.jpg?height=392&width=405&top_left_y=753&top_left_x=905)

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Solution

The area of the larger square is $4 \cdot 216 + 36 = 900 \mathrm{~m}^{2}$. Thus, the side of the larger square is $\sqrt{900} = 30 \mathrm{~m}$. If $x$ is the length of the shorter side of the rectangle, analyzing the larger square we have $2x + 6 = 30$ and thus $x = 12$. Therefore, the dimensions of the sides of the rectangles are $x = 12 \mathrm{~m}$ and $x + 6 = 18 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_09dbad5281c800d666d6g-31.jpg?height=399&width=441&top_left_y=1796&top_left_x=911)