Difference between revisions of "1994 AIME Problems/Problem 8"
RoFlLoLcOpT (talk | contribs) (→Solution) |
(→Solution) |
||
Line 12: | Line 12: | ||
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>. | ||
+ | |||
+ | ''Note'': There is another solution where the point <math>b+37i</math> is a rotation of <math>-60</math> degrees of <math>a+11i</math>; however, this gives <math>a=-21\sqrt 3</math> and <math>b=-5\sqrt 3</math>, which gives the same final answer, <math>315</math>. | ||
== See also == | == See also == |
Revision as of 20:13, 27 February 2011
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 .
Note: There is another solution where the point is a rotation of degrees of ; however, this gives and , which gives the same final answer, .
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 |