2004 AIME I Problems/Problem 4
A square has sides of length 2. Set is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set enclose a region whose area to the nearest hundredth is . Find .
Without loss of generality, let , , , and be the vertices of the square. Suppose the endpoints of the segment lie on the two sides of the square determined by the vertex . Let the two endpoints of the segment have coordinates and . Because the segment has length 2, . Using the midpoint formula, we find that the midpoint of the segment has coordinates . Let be the distance from to . Using the distance formula we see that . Thus the midpoints lying on the sides determined by vertex form a quarter-circle with radius 1. The set of all midpoints forms a quarter circle at each corner of the square. The area enclosed by all of the midpoints is to the nearest hundredth. Thus