Xenie had a rectangle with dimensions $24 \mathrm{~cm}$ and $30 \mathrm{~cm}$. She divided it into several identical rectangular pieces using three segments.

What could be the dimensions of these pieces? Determine four possibilities.

(K. Pazourek)
To ensure the resulting sections are rectangular, the dividing segments must be parallel to the sides of the original rectangle. Depending on the number of segments parallel to the longer or shorter sides of the original rectangle, we can distinguish the following cases:
![](https://cdn.mathpix.com/cropped/2024_04_17_42ed658c189f90810637g-2.jpg?height=250&width=1584&top_left_y=1726&top_left_x=242)

The newly formed rectangles must be identical, meaning each side of the original rectangle must be a multiple of the side of the smaller rectangle. By dividing the dimensions \(24 \, \text{cm}\) and \(30 \, \text{cm}\) according to the above possibilities, we determine the dimensions of the newly formed rectangles:

$$
30 \, \text{cm} \times 6 \, \text{cm}, \quad 15 \, \text{cm} \times 8 \, \text{cm}, \quad 10 \, \text{cm} \times 12 \, \text{cm}, \quad 7.5 \, \text{cm} \times 24 \, \text{cm}.
$$

Evaluation. 1 point for each possibility; 2 points for the analysis, sketches, etc. For solutions containing only images, award a maximum of 3 points.

Note. It is possible to consider more complex divisions where some dividing segments are shorter than the sides of the given rectangle. In our case, however, no additional solutions can be found.