1994 AIME Problem 8

Question

The points

(0,0),

(a,11),

and

(b,37)

are the vertices of an equilateral triangle. Find the value of

ab.

Answer

The vector

(b,37)

is a rotation of

(a,11)

through

60^\circ

or

-60^\circ

Use the second coordinate of the rotation first, then compute the first coordinate

For a

60^\circ

rotation,

37=\frac{\sqrt3}{2}a+\frac{11}{2},

so

a=21\sqrt3.

The first coordinate is then

b=\frac a2-\frac{11\sqrt3}{2}=5\sqrt3.

The opposite orientation changes both signs and leaves the product unchanged. Hence

ab=(21\sqrt3)(5\sqrt3)=315.

Problem 8 in other years