Problem:
The points (0,0),(a,11), and (b,37) are the vertices of an equilateral triangle. Find the value of ab.
Solution:
Let O=(0,0),A=(a,11), and B=(b,37). Note that reflection of the triangle in the y-axis does not change the value of ab. Thus we may assume that the counterclockwise measure of the angle from OA to OB is 60∘.
Let OB=OA=AB=r, and let ∠AOP=α, where P is a point on the positive x-axis. Then ∠BOP=α+60∘. Since
sin(∠BOP)=sin(α+60∘)=sinαcos60∘+cosαsin60∘
we have
r37=r11⋅21+ra⋅23
from which a=213. Similarly
cos(∠BOP)=cos(α+60∘)=cosαcos60∘−sinαsin60∘
which leads to
rb=ra⋅21−r11⋅23
It follows that b=53, so ab=213⋅53=315.
OR
Let O=(0,0),A=(a,11),B=(b,37), and assume that the counterclockwise measure of the angle from OA to OB is 60∘. Regard A and B as the complex numbers a+11i and b+37i, respectively. Since a rotation of 60∘ about the origin is equivalent to multiplication by cos60∘+isin60∘, we have
(a+11i)(cos60∘+isin60∘)=b+37i
Separating the real and imaginary parts yields
a−11311+a3=2b=74
From the second equation we obtain a=213, and then the first yields b=53. Thus ab=315.
