The Dobráks cultivated tulips in a square bed with a side of 6 meters. Later, they added a square terrace with a side of 7 meters to their cottage. One vertex of the terrace was exactly at the center of the tulip bed, and one side of the terrace divided the side of the tulip bed in the ratio $1: 5$.

In what ratio did the second side of the terrace divide the second side of the bed? By how many square meters did the tulip bed decrease due to the construction of the terrace?

(L. Hozová)
The vertices of the tulip bed square are denoted as $A, B, C, D$, the center of this square as $S$, and the intersections of the sides of the square with the sides of the terrace as $E, F$, see the figure.

![](https://cdn.mathpix.com/cropped/2024_04_17_09a048238110e3004b71g-2.jpg?height=591&width=828&top_left_y=1458&top_left_x=614)

When rotating around the center $S$ by integer multiples of the angle $90^{\circ}$, the square $A B C D$ maps onto itself. Consider, for example, a rotation where vertex $A$ maps to vertex $B$, and thus side $D A$ maps to side $A B$. Points $E$ and $F$ lie on these sides, and angle $E S F$ is a right angle, so point $E$ maps to point $F$. Therefore, both sides of the bed are divided by the sides of the terrace in the same ratio, i.e.,

$$
|D E|:|E A|=|A F|:|F B|=1: 5.
$$

Let $G$ and $H$ be the intersections of lines $S E$ and $S F$ with the remaining sides of the square $A B C D$. In the considered rotation, point $B$ maps to point $C$, point $F$ maps to point $G$, etc. In particular, all quadrilaterals $S E A F, S F B G, S G C H$, and $S H D E$ are congruent. These four quadrilaterals form the entire square $A B C D$, whose area is $6 \cdot 6=36\left(\mathrm{~m}^{2}\right)$. The area of each of them is therefore $36: 4=9\left(\mathrm{~m}^{2}\right)$. The construction of the terrace reduced the tulip bed by $9 \mathrm{~m}^{2}$.

Another solution. Using the same notation as above, divide the square $A B C D$ into a grid of smaller squares with side lengths of $1 \mathrm{~m}$ and assume that point $E$ divides side $D A$ in the ratio $1: 5$. In particular, points $E$ and $S$ are grid points, see the figure.

![](https://cdn.mathpix.com/cropped/2024_04_17_09a048238110e3004b71g-3.jpg?height=454&width=466&top_left_y=710&top_left_x=795)

Angle $E S F$ is a right angle precisely when the marked right triangles are congruent, which occurs when $F$ is a grid point dividing side $A B$ in the ratio $1: 5$. Therefore, both sides of the bed are divided by the sides of the terrace in the same ratio.

The area of quadrilateral $S E A F$ is equal to the sum of the areas of the three marked parts, each of which has an area of $3 \mathrm{~m}^{2}$. The construction of the terrace reduced the tulip bed by $9 \mathrm{~m}^{2}$.

Scoring suggestion. 1 point for determining the ratio in which the second side of the terrace divides the second side of the bed; 2 points for the area of the part of the bed occupied by the construction of the terrace; 3 points based on the quality of the commentary.