2018 AIME II Problems/Problem 4
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
See Also
2018 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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