Construct a square $A B C D$ with a side length of $6 \mathrm{~cm}$. Construct a line $p$ parallel to the diagonal $A C$ and passing through point $D$. Construct a rectangle $A C E F$ such that vertices $E$ and $F$ lie on the line $p$.

From the given data, calculate the area of the rectangle $A C E F$.

(L. Růžičková)
Construction:

- Square $A B C D$ with side length $6 \mathrm{~cm}$,
- line $p$ as a parallel to line $A C$ (or perpendicular to line $B D$) passing through point $D$,
- point $E$ as the foot of the perpendicular from point $C$ to line $p$,
- point $F$ as the foot of the perpendicular from point $A$ to line $p$.

![](https://cdn.mathpix.com/cropped/2024_04_17_6fa2a6d202537c5556a1g-2.jpg?height=740&width=828&top_left_y=775&top_left_x=614)

Calculation: The diagonals of a square are equal, perpendicular to each other, and their intersection is the midpoint of both diagonals. The intersection of the diagonals in square $A B C D$ is denoted as $S$.

From the above, it follows that the square $A B C D$ is divided by the diagonals into four congruent triangles $A B S, B C S, C D S$, and $D A S$. Furthermore, the rectangle $A C E F$ is divided by segment $S D$ into two congruent squares, each of which is further divided by a diagonal ($C D$, or $D A$) into two congruent triangles. Thus, both the square $A B C D$ and the rectangle $A C E F$ consist of four congruent triangles, two of which are common to both figures. Therefore, the rectangle $A C E F$ has the same area as the square $A B C D$, which is

$$
6 \cdot 6=36\left(\mathrm{~cm}^{2}\right)
$$

Evaluation. 2 points for performing the construction; 4 points for the calculation, of which 3 points for dividing into congruent triangles and 1 point for determining the area. A calculation based solely on measurement in the image, or a correct result without justification, should not be evaluated.