2009 AIME II Problems/Problem 3
Problem
In rectangle ,
. Let
be the midpoint of
. Given that line
and line
are perpendicular, find the greatest integer less than
.
Solution
![[asy] pair A=(0,10), B=(0,0), C=(14,0), D=(14,10), Q=(0,5); draw (A--B--C--D--cycle); pair E=(7,10); draw (B--E); draw (A--C); pair F=(6.7,6.7); label("\(E\)",E,N); label("\(A\)",A,NW); label("\(B\)",B,SW); label("\(C\)",C,SE); label("\(D\)",D,NE); label("\(F\)",F,W); label("\(100\)",Q,W); [/asy]](http://latex.artofproblemsolving.com/b/d/a/bda5479bee464bcc5f5c02a387f2f7ed6129f333.png)
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
.