Hello, Veronica1999!

Find an example of a regular hexagon whose sides are all $\displaystyle \sqrt{13}$ units long.

Give coordinates for all six vertices.

A regular hexagon is comprised of six equilateral triangles. Code:

C B
* - - - - - *
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
D * - - - - - * - - - - - * A
\ /O\ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
* - - - - - *
E F

Each line segment has length $\displaystyle \sqrt{13}.$

Let the center of the hexagon $\displaystyle O$ be at the Origin.

Then $\displaystyle A$ and $\displaystyle D$ are at: .$\displaystyle \left(\pm\sqrt{13},\:0\right)$

The $\displaystyle x$-coordinate of $\displaystyle B$ is $\displaystyle \frac{1}{2}\sqrt{13}$

The $\displaystyle y$-coordinate of $\displaystyle B$ is the altitude of $\displaystyle \Delta ABO.$

. . which is: .$\displaystyle \left(\frac{1}{2}\sqrt{13}\right)\left(\sqrt{3}\ri ght) \:=\:\frac{1}{2}\sqrt{39}$

Then $\displaystyle B,C,E,F$ are at: .$\displaystyle \left(\pm\frac{1}{2}\sqrt{13},\:\pm\frac{1}{2}\sqr t{39}\right)$