Two semicircles are inscribed in a 50 by 100 rectangle, as shown below. Find R.
The circles will be tangent at the center of the rectangle = (50, 25)
Dropping a perpendicular from this point to the bottom side of the rectangle will form one leg of a right triangle = 25
Connecting the center of the bottom semi-circle to the center of the square is the hypotenuse of this right triangle = R
And the other leg of thisright triangle = 50 - R
So we have
(50-R)^2 + 25^2 = R^2 simplify
R^2 - 100R + 2500 + 625 = R^2
100R = 3125
R = 3125/100 = 125 / 4 = 31.25