Please help ASAP!

The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in the simplest radical form.

The ratio is    √3 / 2 

The diagonals of a regular hexagon have two possible lengths. What is the ratio of the shorter length to the longer length? Express your answer as a common fraction in simplest radical form.

Hello Ericthegg!

$c^2=a^2+b^2-2ab\ cos\ 120^0\\ s^2=2\cdot (\frac{d}{2})^2-2\cdot (\frac{d}{2})^2\cdot (-\frac{1}{2})\\ s^2=2\cdot (\frac{d}{2})^2\cdot(1-(-\frac{1}{2}))\\ s^2=3\cdot (\frac{d}{2})^2$

$s=\frac{d}{2}\sqrt{3}$

The ratio is   $\large \frac{s}{d}=\frac{\sqrt{3}}{2}$

  !

Thank you so much!