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
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From this image, we can see that the oordinate of F is 4, and from this, we can gather that the oordinates 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 . Since is a regular hexagon, . 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 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 |