Difference between revisions of "2004 AIME I Problems/Problem 4"
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
Revision as of 15:00, 27 April 2008
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 
See also
| 2004 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
