Difference between revisions of "1994 AIME Problems/Problem 8"
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Equating the real and imaginary parts, we have: | Equating the real and imaginary parts, we have: | ||
− | <cmath>\begin{align*}b&=a | + | <cmath>\begin{align*}b&=\frac{a}{2}-\frac{11\sqrt{3}}{2}\37&=\frac{11}{2}+\frac{a\sqrt{3}}{2} \end{align*}</cmath> |
Solving this system, we find that <math>a=21\sqrt{3}, b=5\sqrt{3}</math>. Thus, the answer is <math>\boxed{315}</math>. | Solving this system, we find that <math>a=21\sqrt{3}, b=5\sqrt{3}</math>. Thus, the answer is <math>\boxed{315}</math>. |
Revision as of 14:01, 19 July 2009
Problem
The points , , and are the vertices of an equilateral triangle. Find the value of .
Solution
Consider the points on the complex plane. The point is then a rotation of degrees of about the origin, so:
Equating the real and imaginary parts, we have:
Solving this system, we find that . Thus, the answer is .
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |