Ayaev A.V.

On a plane, a regular hexagon is drawn, the length of its side is 1. Using only a ruler, construct a segment whose length is $\sqrt{7}$.

#
Let a regular hexagon $A B C D E F$ with side 1 be given. The diagonal $A C$ is equal to $\sqrt{3}$. Take the point $G$ as the intersection of the lines $C D$ and $E F$. The triangle $D E G$ is equilateral, so $D G=1$. In the right triangle $A C G \quad A C=\sqrt{3}, C G=2$. By the Pythagorean theorem, $A G=\sqrt{7}$.