I might have gone around the world to get this answer but here goes

I let

AP=AS=QC=CR=t

and

PB=BQ=RD=DS=g

I then used algebra to get an expression for the area of the triangles and put it equal to 200

from this I could get t in terms of g

Then you can use pythagoras to find QR squared and QP squared in terms of g

Then you can use pythagoras again to find PR.