Straightedge-only construction of segment of length $\sqrt{7}$, given regular hexagon with unit sides

Question

Let's consider a regular hexagon with unit side length. Draw a line segment of length $\sqrt{7}$ using nothing except a straightedge (that is, an unmarked ruler). The position of the segment may be chosen as you wish.

My thoughts: I've drawn all sorts of lines and tried to get the segment in one of the inside triangles, but to no avail.

Answer 1

The rectangle that has as basis a diagonal of the hexagon and as height the orthogonal median ( joining mean points of opposite sides) has diagonal = $\sqrt 7$.

Added after Blue comment.

All lines can be drown with an unmarked straightedge.

Answer 2

Here's a variation on @Emilio's answer:

Extend two hexagon edges to get $A$; then connect $A$ to $B$.