4. Accurately calculate the lengths of the sides of a rectangle whose perimeter is $4 \mathrm{~cm}$, and the angle between the diagonals is $60^{\circ}$.

$$
\text { (7 points) }
$$
4. We draw a sketch and write down the relationships between the lengths of the sides: $\tan 30^{\circ}=\frac{b}{a}$ and $2 a+2 b=4$.
We express, for example, $b=\frac{\sqrt{3}}{3} a$ and substitute it into $2 a+2 b=4$. We get $2 a+2 \frac{\sqrt{3}}{3} a=4$. We calculate
$a=3-\sqrt{3} \text{ cm}$ and $b=\sqrt{3}-1 \text{ cm}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_97c791425a4f6cf5fca5g-09.jpg?height=345&width=596&top_left_y=1278&top_left_x=730)
Correct sketch with the angle between the diagonals marked ..... 1 point
Writing the relationship $\tan 30^{\circ}=\frac{b}{a}$ ..... 1 point
Writing or using $2 a+2 b=4$ ..... 1 point
Solving the system of equations ..... $2^{*}$ points
Calculating $a=3-\sqrt{3} \text{ cm}$ ..... 1 point
Calculating $b=\sqrt{3}-1 \text{ cm}$ ..... 1 point

NOTE: If the units are missing in the result, deduct 1 point.