1994 AIME Problems/Problem 8
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:
![\[(a+11i)\left(\text{cis}\,60^{\circ}\right) = (a+11i)\left(\frac 12+\frac{\sqrt{3}i}2\right)=b+37i.\]](http://latex.artofproblemsolving.com/f/3/b/f3b0745d80c034402737de449a7bba856fafef5b.png)
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 triangle is just a reflection of the first triangle by the 
-axis, and the signs of 
and 
are flipped. However, the product 
is unchanged.
Solution Two
Using the Pythagorean theorem with these beastly numbers doesn't seem promising. How about properties of equilateral triangles? 
and perpendiculars inspires this solution:
First, drop a perpendicular from 
to 
. Call this midpoint of 
. Thus, 
. The vector from to 
is ![$[\frac{a+b}{2}, 24]$](http://latex.artofproblemsolving.com/b/8/e/b8ecb8d23d37c192caa4555aaacecf5015700c28.png)
. Meanwhile from point we can use a vector with 
the distance; we have to switch the 
and and our displacement is ![$[8\sqrt{3}, \frac{(a+b)\sqrt{3}}{6}]$](http://latex.artofproblemsolving.com/5/4/b/54b45450354b5545f5ca1e5abd63eeedaabb95c3.png)
. (Do you see why we switched and due to the rotation of 90 degrees?)
We see this displacement from to 
is ![$[\frac{a-b}{2}, 13]$](http://latex.artofproblemsolving.com/c/a/6/ca669ebd3feac902552cb45cdb3535e676ee48f7.png)
as well. Equating the two vectors, we get 
and 
. Therefore, 
and 
. And the answer is .
Note: This solution was also present in Titu Andreescu and Zuming Feng's "103 Trigonometry Problems".
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 | ||
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