2016
Problem - 3954
The points $(0,0)$, $(a,11)$, and $(b,37)$ are the vertices of an equilateral triangle. Find the value of $ab$.
Answer
315
One way to solve this problem is to use the vector rotation method. We have $$(a+11i)(\cos 60^{\circ} + i\sin 60^{\circ}) = b + 37i \Leftrightarrow \left(\frac{1}{2}a-\frac{11}{2}\sqrt{3}\right) +\left(\frac{11}{2} +\frac{\sqrt{3}}{2}a\right)i = b + 37i$$
Setting both the real and imaginary parts equal leads to $$(a, b) =(21\sqrt{3}, 5\sqrt{3}) \implies ab = \boxed{315}$$