Difference between revisions of "2009 AIME II Problems/Problem 3"
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+ | In rectangle <math>ABCD</math>, <math>AB=100</math>. Let <math>E</math> be the midpoint of <math>\overline{AD}</math>. Given that line <math>AC</math> and line <math>BE</math> are perpendicular, find the greatest integer less than <math>AD</math>. | ||
+ | == Solution == | ||
<center><asy> | <center><asy> | ||
pair A=(0,10), B=(0,0), C=(14,0), D=(14,10), Q=(0,5); | pair A=(0,10), B=(0,0), C=(14,0), D=(14,10), Q=(0,5); |
Revision as of 20:17, 11 April 2009
Problem
In rectangle , . Let be the midpoint of . Given that line and line are perpendicular, find the greatest integer less than .
Solution
From the problem, and triangle is a right triangle. As is a rectangle, triangles , and are also right triangles. By , , and , so . This gives . and , so , or , so , or , so the answer is .