1994 AIME Problems/Problem 8

Problem

The points $(0,0)\,$ , $(a,11)\,$ , and $(b,37)\,$ are the vertices of an equilateral triangle. Find the value of $ab\,$ .

Using complex coordiantes is one approach. The point $b+37i$ is then a rotation of $60$ degrees of $a+11i$ about the origin, hence we have:

$(a+11i)(cis60)=b+37i \newline(a+11i)(1/2+\sqrt{3}i/2)=b+37i$ .

Setting the real and imaginary parts equal, we have:

$b=a/2-11\sqrt{3}/2 \newline37=11/2+a\sqrt{3}/2$ .

Solving this system, we have: $a=21\sqrt{3}, b=5\sqrt{3}$ . Thus, the answer is $315$ .

1994 AIME (Problems • Answer Key • Resources)
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