Four villages are located at the vertices of a square with a side length of 1 km. To allow travel from each village to any other, two straight roads were laid along the diagonals of the square. Can the road network between the villages be laid out in a different way so that the total length of the roads is reduced, but it is still possible to travel from each village to any other?
Try to find a road network consisting of five straight segments.

## Solution

Let $A, B, C, D$ be the villages at the vertices of the square. On the segment connecting the midpoints of sides $A B$ and $C D$, take points $M$ and $N$, which are $3 / 8$ away from sides $A B$ and $C D$. Then $A M=B M=C N=D N=5 / 8, M N=1 / 4$. Thus, the length of the new road network is 2.75, while the length of the old network is $2 \sqrt{2}>2.8$.

## Answer

It is possible.