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This problem trains for: SAT-I, SAT-II, AMC-8.

A circle of radius R intersects a set of 5 equidistant parallel lines as in the figure:

figure

The line AF is such that it intersects both the circle and the line L2 at A and the circle and the line L1 at F.

What is the maximum possible length of the segment AB denoted by x?

The longest chord of any circle is a diameter. If the segment AF passes through the center of the circle, i.e. if it is a diameter, then its length is maximized.

The segments determined by equidistant parallel lines on a secant (transverse) are congruent. Therefore, the longest possible segment AB is equal to half of the radius of the circle.