1994 AIME Problems/Problem 8

Revision as of 19:26, 20 June 2008 by Dgreenb801 (talk | contribs)

Problem

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

Solution

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 $315$.

See also

1994 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
Invalid username
Login to AoPS