1999 AIME Problems/Problem 2
Contents
Problem
Consider the parallelogram with vertices ,
,
, and
. A line through the origin cuts this figure into two congruent polygons. The slope of the line is
where
and
are relatively prime positive integers. Find
.
Solution 1
Let the first point on the line be
where a is the height above
. Let the second point on the line
be
. For two given points, the line will pass the origin iff the coordinates are proportional (such that
). Then, we can write that
. Solving for
yields that
, so
. The slope of the line (since it passes through the origin) is
, and the solution is
.
Solution 2
You can clearly see that a line that cuts a parallelogram into two congruent pieces must go through the center of the parallelogram. Taking the midpoint of , and
gives
, which is the center of the parallelogram. Thus the slope of the line must be
, and the solution is
.
See also
1999 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |