In the dodecagon $A B C D E F G H I J K L$, every two adjacent sides are perpendicular, and all sides except for sides $A L$ and $G F$ are congruent. Sides $A L$ and $G F$ are twice as long as the other sides. Segments $B G$ and $E L$ intersect at point $M$. The quadrilateral $A B M J$ has an area of $1.8 \mathrm{~cm}^{2}$.

Determine the area of the quadrilateral $E F G M$.

(E. Semerádová)

![](https://cdn.mathpix.com/cropped/2024_04_17_848c3145926e37c65b12g-1.jpg?height=503&width=483&top_left_y=1936&top_left_x=792)

Hint. Divide the image into smaller congruent shapes.
From the problem statement, we know that the dodecagon $A B C D E F G H I J K L$ consists of six identical squares. Each of these squares can be further divided by diagonals into four identical triangles. Therefore, the dodecagon consists of 24 identical triangles, and each of the above-mentioned figures is formed by several such triangles:

![](https://cdn.mathpix.com/cropped/2024_04_17_848c3145926e37c65b12g-2.jpg?height=503&width=483&top_left_y=565&top_left_x=792)

It is now clear that the areas of quadrilateral $A B M J$ and quadrilateral $E F G M$ are in the ratio $3: 7$. Since the area of quadrilateral $A B M J$ is $1.8 \mathrm{~cm}^{2}$, the area of quadrilateral $E F G M$ is $4.2 \mathrm{~cm}^{2}$.