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
Solution 1
![[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
.
Solution 2
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
.
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |