Problem:
A square has sides of length . Set is the set of all line segments that have length 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 .
Solution:
Let be a line segment in set that is not a side of the square, and let be the midpoint of . Let be the vertex of the square that is on both the side that contains and the side that contains . Because is the median to the hypotenuse of right . Thus every midpoint is unit from a vertex of the square, and the set of all the midpoints forms four quarter-circles of radius and with centers at the vertices of the square. The area of the region bounded by the four arcs is , so .
Place a coordinate system so that the vertices of the square are at , , and . When the segment's vertices are on the sides that contain , its endpoints' coordinates can be represented as and . Let the coordinates of the midpoint of the segment be . Then and . Thus , and the midpoints of these segments form a quarter-circle with radius centered at the origin. The set of all the midpoints forms four quarter-circles, and the area of the region bounded by the four arcs is , so .
The problems on this page are the property of the MAA's American Mathematics Competitions