Construct a square $A B C D$ with a side length of $6 \text{~cm}$ and mark the intersection of its diagonals as $S$. Construct point $K$ such that together with points $S, B, C$ it forms a square $B K C S$. Construct point $L$ such that together with points $S, A, D$ it forms a square $A S D L$. Construct the segment $K L$, mark the intersection of segments $K L$ and $A D$ as $X$, and the intersection of segments $K L$ and $B C$ as $Y$.

From the given data, calculate the length of the broken line $K Y B A X L$. (L. Růžičková)
Construction:

- Square $A B C D$ with side length $6 \mathrm{~cm}$,
- point $S$ as the intersection of segments $A C$ and $B D$,
- point $K$ as the intersection of the perpendicular line $\mathrm{k} S B$ at point $B$ and the perpendicular line to $S C$ at point $C$,
- point $L$ as the intersection of the perpendicular line $\mathrm{k} S A$ at point $A$ and the perpendicular line $\mathrm{k} S D$ at point $D$,
- points $X$ and $Y$ as the intersections of segment $K L$ with sides $A D$ and $B C$.

Calculation: The length of the broken line $K Y B A X L$ is the sum of the lengths of segments $K Y, Y B, B A, A X$, and $X L$. The length of segment $A B$ is $6 \mathrm{~cm}$.

Point $Y$ is the intersection of the diagonals of square $B K C S$, so it is the midpoint of each of these segments. Segment $B C$ is a side of the given square, and it measures $6 \mathrm{~cm}$. Therefore, each of the segments $K Y$ and $Y B$ measures $3 \mathrm{~cm}$.

For the same reason, each of the segments $A X$ and $X L$ measures $3 \mathrm{~cm}$.

The length of the broken line $K Y B A X L$ is $3+3+6+3+3=18(\mathrm{~cm})$.

![](https://cdn.mathpix.com/cropped/2024_04_17_23afa01ae7ab6ca1f7edg-3.jpg?height=585&width=1054&top_left_y=316&top_left_x=501)

Evaluation. 2 points for performing the construction. 4 points for the calculation, of which 2 points for determining and justifying that points $X$ and $Y$ are the midpoints of the respective squares, and 2 points for solving and the result. For simply adding the lengths of the segments without justification (or by referring to the measured lengths in the constructed diagram), award no more than 1 point.