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 60β.
Let OB=OA=AB=r, and let β AOP=Ξ±, where P is a point on the positive x-axis. Then β BOP=Ξ±+60β. Since
It follows that b=53β, so ab=213ββ 53β=315β.
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 60β. Regard A and B as the complex numbers a+11i and b+37i, respectively. Since a rotation of 60β about the origin is equivalent to multiplication by cos60β+isin60β, we have
(a+11i)(cos60β+isin60β)=b+37i
Separating the real and imaginary parts yields
aβ113β11+a3ββ=2b=74β
From the second equation we obtain a=213β, and then the first yields b=53β. Thus ab=315β.
