Problem 1 Complex Numbers

Problem 1: Complex Numbers

AIME 1994 #8

Problem: The points (0, 0), (a, 11), and (b, 37) are the vertices of an equilateral triangle. Find the value of ab. Lemma: The rotation of the point (x, y) counterclockwise about an angle of θ is given by x0 = x cos θ − y sin θ y 0 = x sin θ + y cos θ Corollary: The rotation of the point x + yi counterclockwise about an angle of θ is given by (x + yi)(cos θ + i sin θ)

Solution: Consider the equilateral triangle in the complex plane. That is, (a, 11) represents the point a+11i and (b, 37) represents the point b+37i. Then b+37i is a 60◦ counterclockwise rotation of a + 11i about the origin, or (a + 11i)cis(60◦ ) = b + 37i ! √ 3i 1 (a + 11i) + = b + 37i 2 2 √ √ a 3ai 11i 11 3 + + − = b + 37i 2 2 2 2 Setting the real and imaginary parts equal: √ a − 11 3 = 2b

√

3a + 11 = 74

√ √ √ 63 From the second equation, a = √ = 21 3. Then we obtain 21 3 − 11 3 = 2b or 3 √ b = 5 3 after plugging back into the first equation. √
√
Finally, ab = 21 3 5 3 = 315 .

Solution was written by Qiaochu Yuan and compiled from Art of Problem Solving Forums.