2019 AMC 12B Problems/Problem 24
Contents
[hide]Problem
Let Let denote all points in the complex plane of the form where and What is the area of ?
Solution
Let be the third root of unity. We wish to find the span of for reals . Note that if , then forms the same point as . Therefore, assume that at least one of them is equal to . If only one of them is equal to zero, we can form an equilateral triangle with the remaining two, of side length . Similarly for if two are equal to zero. So the area of the six equilateral triangles is
Here is a diagram:
-programjames1
Solution 2
We can add on each term one at a time. First off, the possible values of lie on the following graph:
For each point on the line, we can add . This means that we can extend the area to
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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