The four cities $A, B, C$ and $D$ were built along a straight highway, as illustrated.

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-002.jpg?height=72&width=397&top_left_y=889&top_left_x=1429)

The distance between $A$ and $C$ is $50 \mathrm{~km}$ and the distance between $B$ and $D$ is $45 \mathrm{~km}$. Additionally, it is known that the distance between the first and the last city is $80 \mathrm{~km}$. What is the distance, in kilometers, between the cities $B$ and $C$?
(a) 15
(c) 20
(c) 25
(d) 5
(e) 10
The correct option is (a).

Solution 1: From the problem statement, we have $A C=50, B D=45$ and $A D=80$. From the figure, it follows that

$$
C D=A D-A C=80-50=30
$$

Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_01_666a1aec7e447b0a6cbeg-036.jpg?height=262&width=503&top_left_y=1274&top_left_x=1379)

$$
B C=B D-C D=45-30=15 \mathrm{~km}
$$

Solution 2: From the figure, we have that $45-B C=80-50$. Thus, $B C=15 \mathrm{~km}$.