From point $A$ to point $C$, there is an educational trail passing through point $B$, and there is also a red tourist trail, see the image. In addition, a shortcut not shown in the image can be used, which is 1500 meters long, starting at $A$ and ending on the educational trail. Vojtěch found that

1. the trip from $A$ along the red trail to $C$ and back along the educational trail to $A$ is 7700 meters long,
2. the trip from $B$ along the educational trail to $C$ and then along the red trail to $A$ is 5800 meters long,
3. using the shortcut, the trip from $A$ to $B$ is 1700 meters long,
4. the trip from $A$ along the educational trail to $C$ and back to $A$ first along the educational trail and then along the shortcut is 8800 meters long.

Determine the length of the educational trail from $A$ to $C$. If the problem allows multiple answers, provide all of them.

(L. Simünek)

![](https://cdn.mathpix.com/cropped/2024_04_17_5752a8a8cd9a6cffa1f9g-2.jpg?height=556&width=552&top_left_y=967&top_left_x=752)

Hint. Determine where the shortcut could end on the educational trail.
The shortcut could have led to the educational trail either in the section between $A$ and $B$, or in the section between $B$ and $C$.

![](https://cdn.mathpix.com/cropped/2024_04_17_5752a8a8cd9a6cffa1f9g-2.jpg?height=565&width=548&top_left_y=1788&top_left_x=754)

The shortcut was $1500 \mathrm{~m}$ long, and with its use, the journey from $A$ to $B$ (according to the 3rd piece of information) was $1700 \mathrm{~m}$. The distance from $B$ to the mouth of the shortcut was therefore the same in both cases, and that is

$$
1700-1500=200(\mathrm{~m}) .
$$

From the 1st and 2nd pieces of information, it follows that the journey from $A$ to $B$ along the educational trail was

$$
7700-5800=1900(\mathrm{~m}) .
$$

If we denote the length of the section of the educational trail between $B$ and $C$ as $y$, then using the 4th piece of information, we get two different equations depending on where the shortcut leads:

a) For the mouth in the section between $A$ and $B$, we have

$$
\begin{aligned}
1900+2 y+200+1500 & =8800, \\
2 y & =5200, \\
y & =2600(\mathrm{~m}) .
\end{aligned}
$$

In this case, the entire educational trail was

$$
1900+2600=4500(\mathrm{~m}) \text {. }
$$

b) For the mouth in the section between $B$ and $C$, we have

$$
\begin{aligned}
1900+2 y-200+1500 & =8800 \\
2 y & =5600 \\
y & =2800(\mathrm{~m})
\end{aligned}
$$

In this case, the entire educational trail was

$$
1900+2800=4700(\mathrm{~m}) \text {. }
$$

Note. If we denote the length of the section of the educational trail between $A$ and $B$ as $x$, between $B$ and the mouth of the shortcut as $z$, and the length of the red path between $A$ and $C$ as $w$, then the information from the problem can be written as equations as follows:

1. $w+y+x=7700$,
2. $y+w=5800$,
3. $1500+z=1700$,
4. $x+2 y+z+1500=8800$, or $x+2 y-z+1500=8800$.

In the previous, a step-by-step solution of this system for the unknowns $z, x$, and $y$ (and the subsequent expression of the sum $x+y$) is presented. From the second equation, the value of the unknown $w$ can also be expressed.