2009 AIME II Problems/Problem 3
In rectangle , . Let be the midpoint of . Given that line and line are perpendicular, find the greatest integer less than .
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 .
Let be the ratio of to . On the coordinate plane, plot , , , and . Then . Furthermore, the slope of is and the slope of is . They are perpendicular, so they multiply to , that is, which implies that or . Therefore so .
Similarly to Solution 2, let the positive x-axis be in the direction of ray and let the positive y-axis be in the direction of ray . Thus, the vector and the vector are perpendicular and thus have a dot product of 0. Thus, calculating the dot product:
Substituting AD/2 for x:
Draw and to form a parallelogram . Since , by the problem statement, so is right. Letting , we have and . Since , . Solving this, we have , so the answer is .
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