size of the biggest possible square that fits into a circle segment

Question

I currently have an exercise that involves putting the biggest possible square into a circle segment with an angle of 30 degrees and a lenght of 1.
I was able to draw a sketch and almost solve it, but theres just a tiny little part of the segment (z) which I just cannot find out how to calculate. I was able to create the following equation: x=1-2x*sqrt(3/4)-z, its just that I don't know how to calculate z. Can somebody help me out? Heres my sketch:

Answer 1

Pythagoras solves it for you. There is a right-angled triangle in your diagram whose sides are $1,x,$ and $x+2x\sqrt\frac34$.