The length of the arc can be found using the formula
Arc = 2πr*θ/360
where r is the radius of the circle, and θ is an angle of the sector.
Applying this formula in this problem, we can get the ratio of DE to BC as
(2*π(r/2)*β/360)/ (2πr*2β/360) = πr/(4πr)
Thus, answer A is correct.