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.
Well, let's see what we know. We know that the diagonals of a hexagon are the lines connecting any two non-adjacent vertices. That's right. And we know that a regular hexagon is a hexagon with all equal sides. Yes, so we can call the side length x. I see. So we can use the height of one of those equilateral triangles to find the length of the diagonals. Exactly. The height of an equilateral triangle is $\frac{s\sqrt{3}}{2}$. And we want to find the length of the diagonal in terms of s. Yes. So we can call the length of the height h. So $h=\frac{\sqrt{3}}{2}s$. That's right. So one diagonal is $2s$. And the other diagonal corresponds to twice the height of the equilateral triangle. So, the other diagonal is $2h$. And we know that $h=\frac{\sqrt{3}}{2}s$. So the other diagonal is $\sqrt{3}s$. Which simplifies to $\sqrt{3}s$. Simplifying gives $\frac{\sqrt{3}}{2}$ And that is our final answer.

