Mr. Kostkorád owned a rectangular garden, on which he gradually paved paths from one side to the other. The paths were of the same width, intersected at two points, and once a paved area was created, it was skipped during further paving.

When Mr. Kostkorád paved a path parallel to the longer side, he used $228 \mathrm{~m}^{2}$ of paving stones. Then he paved a path parallel to the shorter side and used $117 \mathrm{~m}^{2}$ of paving stones.

Finally, he paved another path parallel to the first path, this time using only $219 \mathrm{~m}^{2}$ of paving stones.

Determine the dimensions of Mr. Kostkorád's garden.

(M. Petrová)

Hint. Why was less paving used for the third path than for the first one?
Let's outline how Kostkorád's garden looked (numbers indicate the order of the paved paths):

![](https://cdn.mathpix.com/cropped/2024_04_17_adee0188cf1f8dcaeca5g-2.jpg?height=460&width=786&top_left_y=587&top_left_x=635)

The first and third paths have the same dimensions, and thus the same area. Less paving was used for the third path compared to the first because the third path was not paved where it intersects with the second path (it was already paved there). The common part of the second and third paths is a square whose side length corresponds to the width of the path. The paving used for this square is

$$
228-219=9\left(\mathrm{~m}^{2}\right)
$$

A square with an area of $9 \mathrm{~m}^{2}$ has a side length of $3 \mathrm{~m}$, so the width of all three paths is $3 \mathrm{~m}$. From this and the amount of paving used for each path, we can determine the dimensions of the garden:

For the first path, $228 \mathrm{~m}^{2}$ of paving was used, which is the actual area of the path. The length of the garden is

$$
228: 3=76(\mathrm{~m}) \text {. }
$$

For the second path, $117 \mathrm{~m}^{2}$ of paving was used, which is $9 \mathrm{~m}^{2}$ less than the actual area of the path (the common square with the first path was already paved). The width of the garden is

$$
(117+9): 3=126: 3=42(\mathrm{~m}) \text {. }
$$

The dimensions of Kostkorád's garden are $76 \mathrm{~m}$ and $42 \mathrm{~m}$.

Note. If the solver does not add those $9 \mathrm{~m}^{2}$ when calculating the width of the garden, they will receive an incorrect dimension of $39 \mathrm{~m}$. Such a solution should be rated no higher than "good."