5.3. One side of a rectangle was increased by 3 times, and the other was reduced by 2 times, resulting in a square. What is the side of the square if the area of the rectangle is $54 \mathrm{m}^{2} ?$
Answer: 9 m.

Let's reduce the side of the given rectangle by half (see Fig. 5.3a). Then the area of the resulting rectangle will be 27 m $^{2}$ (see Fig. 5.3b). Next, we will increase the other side by three times, that is, "add" two more rectangles (see Fig. 5.3c). The area of the resulting figure will become $27 \cdot 3=81$ (m ${ }^{2}$ ). Since, by the condition, a square has formed, its area is $81 \mathrm{~m}^{2}$. Therefore, the side of the resulting square is 9 meters.

![](https://cdn.mathpix.com/cropped/2024_05_06_413311938b0299e6ad35g-1.jpg?height=75&width=375&top_left_y=1820&top_left_x=320)

Fig. 5.3a

![](https://cdn.mathpix.com/cropped/2024_05_06_413311938b0299e6ad35g-1.jpg?height=78&width=210&top_left_y=1818&top_left_x=1008)

Fig. 5.3b

![](https://cdn.mathpix.com/cropped/2024_05_06_413311938b0299e6ad35g-1.jpg?height=199&width=196&top_left_y=1804&top_left_x=1543)

Fig. 5.3c

+ complete justified solution

$\pm$ the area of the square is correctly obtained, but the side is not found or is found incorrectly

Ғ only the correct answer is provided or the correct answer is obtained by considering a specific case (a rectangle with specific sides)