2000 AIME I Problems/Problem 2
Let and be integers satisfying . Let , let be the reflection of across the line , let be the reflection of across the y-axis, let be the reflection of across the x-axis, and let be the reflection of across the y-axis. The area of pentagon is . Find .
Since , we can find the coordinates of the other points: , , , . If we graph those points, we notice that since the latter four points are all reflected across the x/y-axis, they form a rectangle, and is a triangle. The area of is and the area of is . Adding these together, we get . Since are positive, , and by matching factors we get either or . Since the latter case is the answer, and .
We find the coordinates like in the solution above: , , , , . Then we apply the Shoelace Theorem.
This means that or , but since , then the answer is
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