4. The convex hexagon $A B C D E F$ is composed of an isosceles trapezoid $A C D F$ and two isosceles triangles $A B C$ and $F D E$ with equal heights ($h=12 \mathrm{~cm}$). If $A B=15 \mathrm{~cm}, A F=25 \mathrm{~cm}$ and $F E=20 \mathrm{~cm}$, construct the hexagon at a scale of 1:5 and calculate its area.
Solution. By applying the Pythagorean theorem, it is easily obtained that

$$
A G=9 \mathrm{~cm}, F H=16 \mathrm{~cm}
$$

Now, $F L=7 \mathrm{~cm}$, so again from the Pythagorean theorem it follows that $A L=24 \mathrm{~cm}$.

According to this, the area of the given hexagon is equal to

$$
\begin{aligned}
P & =P_{A B C}+P_{D F A C}+P_{D F E} \\
& =9 \cdot 12+16 \cdot 12+(16+9) \cdot 24 \\
& =900 \mathrm{~cm}^{2}
\end{aligned}
$$

To the reader, according to the given and obtained data, we leave it to you to construct the hexagon yourself.

![](https://cdn.mathpix.com/cropped/2024_06_05_e4a4e8557567895ef107g-5.jpg?height=572&width=437&top_left_y=1610&top_left_x=1029)