Let x be the length of the tether.
Let O be the center of the circle, P the point at which the tether
attaches, and A and B the points on the circle that are x away from P.
The goat can reach the combination of two regions:
the part of the circle centered at O with radius 10 that is outside chord AB
and the part of the circle centered at P with radius x, again outside AB.
Consider a circle of radius r and a chord subtended by a central angle of 2
* c.
The area inside the circle and outside the chord is
r2 * c - r2 * sin c * cos c.
Let t be the measure of angle POA and let g be the measure of angle APO.
By the law of cosines,
x2 = 102 + 102 - 2 * 10 * 10 * cos t and
102 = 102 + x2 - 2 * 10 * x * cos g.
Combining all this information with the fact that the goat should be able to graze on half the field,
we get a mildly messy equation in x.
We can't solve it exactly, but we can approximate the solution as closely
as we like.
To two decimal places, the tether should be 11.59 feet long.