2008 AMC 12A Problems/Problem 23
Problem
The solutions of the equation are the vertices of a convex polygon in the complex plane. What is the area of the polygon?
Solution
Looking at the coefficients, we are immediately reminded of the binomial expansion of .
Modifying this slightly, we can write the given equation as: We can apply a translation of and a rotation of (both operations preserve area) to simplify the problem:
Because the roots of this equation are created by rotating radians successively about the origin, the quadrilateral is a square.
We know that half the diagonal length of the square is
Therefore, the area of the square is
See Also
2008 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
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All AMC 12 Problems and Solutions |
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