The question came up while messing around with graph paper. I wanted to make an isosceles triangle where the length of one side and it's hight were both integers.
The closest I could get was a base side of $8$ and a height of $7$. Which gives a length of the other two sides $\sqrt{65}$. It looks like the ratio of half the bast to the height is $\sqrt{3}$ if that helps.