2018 AIME II Problems/Problem 4
Contents
Problem
In equiangular octagon , and . The self-intersecting octagon encloses six non-overlapping triangular regions. Let be the area enclosed by , that is, the total area of the six triangular regions. Then , where and are relatively prime positive integers. Find .
Solution
We can draw and introduce some points.
The diagram is essentially a 3x3 grid where each of the 9 squares making up the grid have a side length of 1.
In order to find the area of , we need to find 4 times the area of and 2 times the area of .
Using similar triangles and (We look at their heights), . Therefore, the area of is
Since and , and .
Therefore, the area of is
Our final answer is
Solution 2
is essentially a plus sign with side length 1 with a few diagonals, which motivates us to coordinate bash. We let and . To find 's self intersections, we take
And plug them in to get where is the intersection of and , and is the intersection of and .
We also track the intersection of and to get .
By vertical symmetry, the other 2 points of intersection should have the same x-coordinates. We can then proceed with Solution 1 to calculate the area of the triangle (compare the -coordinates of and and ).
See Also
2018 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.