2008 AMC 12A Problems/Problem 23
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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 • Resources) | ||
| Preceded by Problem 22 | Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
