1994 AIME Problems/Problem 8
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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 triangle is just a reflection of the first triangle by the
-axis, and the signs of
and
are flipped. However, the product
is unchanged.
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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