In a sports area, there is a rectangular plot $A B C D$ with the longer side $A B$. The diagonals $A C$ and $B D$ form an angle of $60^{\circ}$. Runners train on the large circuit $A C B D A$ or on the small track $A D A$. Mojmír ran ten times around the large circuit, and Vojta ran fifteen times around the small track, i.e., fifteen times in one direction and fifteen times in the opposite direction. Together, they ran $4.5 \mathrm{~km}$. How long is the diagonal $A C$?

(L. Hozová)

Hint. Is there any relationship between the length of the diagonal and the length of the shorter side of the rectangle?
The intersection of the diagonals is denoted by $S$. We need to decide whether the given angle of $60^{\circ}$ refers to angle $A S B$ or $A S D$. Since for the sides of the rectangle, $|A B|>|A D|$, angle $A S B$ must be obtuse and angle $A S D$ must be acute. Therefore, the measure $60^{\circ}$ belongs to angle $A S D$.

![](https://cdn.mathpix.com/cropped/2024_04_17_47cbfa10f45526792f91g-5.jpg?height=343&width=460&top_left_y=2144&top_left_x=798)

In every rectangle, triangles $A S D$ and $B S C$ are isosceles and congruent to each other. In our case, they are even equilateral. This means that segments $A S, S C, C B, B S, S D$, and $D A$ are congruent, and we denote their length (in meters) by $s$. We want to determine the length of the diagonal, which in the current notation corresponds to the value $2 s$.

The length of the large circuit is thus $6 s$, and the total distance Mojmír ran is $10 \cdot 6 s = 60 s$. The length of the small circuit is $2 s$, and the total distance Vojta ran is $15 \cdot 2 s = 30 s$. Together, they ran $60 s + 30 s = 90 s$, which, according to the problem, is equal to $4.5 \text{ km} = 4500 \text{ m}$. Therefore,

$$
90 s = 4500
$$

from which it follows that $2 s = 100$. The length of the diagonal is $100 \text{ m}$.