For those who have traditionally had difficulty with "word problems". :)
 
We have a goat (g) tethered with a rope (r) to the _edge_ of a circle
(x) at point (t), _not_ the center.  The goal is to find the length of
the rope (the radius of circle (a)) such that the intersection of
circle (x) and circle (a) describes an area equal to half the area of
circle (x).  (I don't have the original text, so this is as close as
it gets.)
 
Express the length of (r) in terms of the radius/diameter of the
circle (x).
 
                    xx          a
               xxx      xxx a
                         a
         xx            a       xx
 
      x             a              x        (No comments about
                                             my circles, please)
   x              a                   x
 
 x                                      x
                a            /r\
x               g------------------------t
                a
 x                                      x
 
   x              a                   x
 
      x             a              x
 
         xx            a       xx
                         a
               xxx      xxx a
                    xx          a
 
If the rope were attached to the center of (x), the answer would be
radius(x)/sqrt(2) as has been pointed out by others.  Which was also
my answer (sort of) until I actually read the problem.
 
By visual inspection, the only thing I can tell for sure is that
radius(a) must be greater than radius(x) and less than the diameter of
(x).  Beyond that profound observation, I retire.
 
Show your work.
Don't look at anyone else's paper.
You may start .... now.
 
------------
Randy Medd
Telamon, Inc.