Inside the rectangle $A B G H$ are two identical squares $C D E F$ and $I J K L$. The side $C F$ of the first square lies on the side $B G$ of the rectangle, and the side $I L$ of the second square lies on the side $H A$ of the rectangle. The perimeter of the octagon $A B C D E F G H$ is $48 \, \text{cm}$, and the perimeter of the dodecagon $A B C D E F G H I J K L$ is $58 \, \text{cm}$. This dodecagon is symmetric with respect to a horizontal axis, and the lengths of all its sides are expressed in whole numbers of centimeters.

Determine the lengths of the sides of the rectangle $A B G H$.

(L. Šimůnek)

![](https://cdn.mathpix.com/cropped/2024_04_17_93ce04c5894139b12786g-1.jpg?height=382&width=600&top_left_y=2059&top_left_x=731)
Significant parts of the perimeters of the octagon $A B C D E F G H$ and the dodecagon $A B C D E F G H I J K L$ are common to both figures. On the other hand, the segments $I J, J K$, and $K L$ are part of only the perimeter of the dodecagon, and the segment $I L$ is part of only the perimeter of the octagon. These four segments are congruent to each other, and their length is denoted by $a$. The difference in the perimeters of the figures, $58-48=10(\mathrm{~cm})$, corresponds to two $a$; i.e., $a=5 \mathrm{~cm}$.

Similarly, the difference between the perimeter of the octagon $A B C D E F G H$ and the perimeter of the rectangle $A B G H$ is equal to two $a$, or $10 \mathrm{~cm}$. Therefore, the perimeter of the rectangle is $48-10=38(\mathrm{~cm})$, and the sum of the lengths of its sides is $38: 2=19(\mathrm{~cm})$.

Given that the figure is symmetric with respect to a horizontal axis, the segments $A L, B C, F G$, and $I H$ must be congruent. Since the lengths of all sides of the dodecagon are expressed in whole numbers in centimeters, the lengths of the sides of the rectangle $A B G H$ can only take the following values (in cm):

$$
\begin{aligned}
|A B| & =11,12,13, \ldots \\
|B G| & =7,9,11, \ldots
\end{aligned}
$$

For the sum of these lengths to be 19, the lengths of the sides of the rectangle $A B G H$ must be $12 \mathrm{~cm}$ and $7 \mathrm{~cm}$.

Note. If the length of the congruent segments $A L, B C, F G$, and $I H$ is denoted by $b$ and the length of the congruent segments $K D$ and $E J$ is denoted by $c$, then the perimeter of the dodecagon can be expressed as follows:

$$
\begin{aligned}
10 a+4 b+2 c & =58 \\
4 b+2 c & =8 .
\end{aligned}
$$

Since the lengths of all sides of the dodecagon (in cm) are integers, $b$ and $c$ must also be integers and positive. Therefore, the problem has a unique solution: $b=1$ and $c=2$. The lengths of the sides of the rectangle $A B G H$ are $2 a+c=12(\mathrm{~cm})$ and $2 b+a=7(\mathrm{~cm})$.

![](https://cdn.mathpix.com/cropped/2024_04_17_93ce04c5894139b12786g-2.jpg?height=411&width=617&top_left_y=1556&top_left_x=728)

Scoring suggestion. 3 points for the length of the side of the square $a$ (1 point for justification); 2 points for the relationship $|A B|+|B G|=19 \mathrm{~cm}$ or its equivalent (e.g., $4 b+2 c=8$); 1 point for calculation and correct conclusion.