"The points $(0,0),\;(a, 11), \text{ and } (b,37)$ are vertices of an equilateral triangle. Find the product $ab$."
I'm not sure how to start this problem. I of course drew out an equilateral triangle with those points, but i'm not sure what information I can draw from them. We know that the sides all have equal length obviously. I'm not sure what to do.
Best Answer
Use complex numbers. Notice that: $$b+37i=(a+11i)e^{i\pi/3} \Rightarrow b+37i=\frac{1}{2}(1+\sqrt{3}i)(a+11i)$$ Comparing the imaginary parts: $$37=\frac{1}{2}(\sqrt{3}a+11) \Rightarrow a=21\sqrt{3}$$ Comparing the real part: $$b=\frac{1}{2}(a-11\sqrt{3})\Rightarrow b=5\sqrt{3}$$ Hence, $$\boxed{ab=315}$$