Difference between revisions of "2019 AMC 12B Problems/Problem 24"
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==Problem== | ==Problem== | ||
| + | Let <math>\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.</math> Let <math>S</math> denote all points in the complex plane of the form <math>a+b\omega+c\omega^2,</math> where <math>0\leq a \leq 1,0\leq b\leq 1,</math> and <math>0\leq c\leq 1.</math> What is the area of <math>S</math>? | ||
| + | <cmath>\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi</cmath> | ||
==Solution== | ==Solution== | ||
Revision as of 12:10, 14 February 2019
Problem
Let 
Let 
denote all points in the complex plane of the form 
where 
and 
What is the area of ?
![\[\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi\]](http://latex.artofproblemsolving.com/8/2/f/82f3c26c24494adee9c3eac633718aa70f5abcc4.png)
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
![\[\boxed{\text{(C) }\frac{3\sqrt{3}}{2}}\]](http://latex.artofproblemsolving.com/5/a/1/5a163e3683b65af361716346b4c015a1ea7950fc.png)
Here is a diagram:
![[asy] size(200,200); draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle); draw((0,0)--(1/2,sqrt(3)/2)--(-1/2,sqrt(3)/2)--cycle); draw((0,0)--(-1/2,sqrt(3)/2)--(-1,0)--cycle); draw((0,0)--(-1,0)--(-1/2,-sqrt(3)/2)--cycle); draw((0,0)--(-1/2,-sqrt(3)/2)--(1/2,-sqrt(3)/2)--cycle); draw((0,0)--(1/2,-sqrt(3)/2)--(1,0)--cycle); draw((-2,0)--(2,0)); draw((0,-2)--(0,2)); [/asy]](http://latex.artofproblemsolving.com/4/2/d/42dee9da6ce367a03187683e00b3061794ee81a5.png)
-programjames1
See Also
| 2019 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 23 |
Followed by Problem 25 |
| 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 | |
| All AMC 12 Problems and Solutions | |
