2004 AIME I Problems/Problem 4
Problem
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
.
Solution
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