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A goat in a tether

I would like to solve the following goat problem in sage.

A goat is on the tether on the boundary of the circle with radius 10. What is the radius of the tether is the goat manages to eat half of the grass?

I managed to solve it by Python when I derived the equation.

import numpy
import scipy
import scipy.optimize
def f(x):
    return y

angle = scipy.optimize.newton(f, 1)

But I was wondering if this can be solved by sage such that I make equations for circle with center (0,0) and radius 10 and circle with center (0,1) and radius x_1, compute the intersections numerically, find the area between curves numerically and iterate this for different radius x_n until the suitable accuracy is found.

The problem, I found is that I don't know how to fund numerically the coordinates of the intersections of two circles and pass them to the numerical integrator.