2003 AIME II Problems/Problem 14
Problem
Let and
be points on the coordinate plane. Let
be a convex equilateral hexagon such that
and the y-coordinates of its vertices are distinct elements of the set
The area of the hexagon can be written in the form
where
and
are positive integers and n is not divisible by the square of any prime. Find
Solution
The y-coordinate of must be
. All other cases yield non-convex hexagons, which violate the problem statement.
Letting , and knowing that
, we can use rewrite
using complex numbers:
. We solve for
and
and find that
and that
.
The area of the hexagon can then be found as the sum of the areas of two congruent triangles ( and
, with height
and base
and a parallelogram (
, with height
and base
.
.
Thus, .
Solution (Incomplete/incorrect)
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
From this image, we can see that the y-coordinate of F is 4, and from this, we can gather that the coordinates of E, D, and C, respectively, are 8, 10, and 6.
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
In this image, we have drawn perpendiculars to the -axis from F and B, and have labeled the angle between the
-axis and segment
. Thus, the angle between the
-axis and segment
is
so,
. Expanding, we have

Isolating we see that
, or
. Using the fact that
, we have
, or
. Letting the side length of the hexagon be
, we have
. After simplification we see that
.
The following is incorrect as the hexagon is NOT regular (although it is equilateral). The previous work IS correct, so I am leaving it as part of an incomplete solution
The area of the hexagon is , so the area of the hexagon is
, or
.
See also
2003 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |