1997 AIME Problems/Problem 15
Contents
Problem
The sides of rectangle 
have lengths 
and 
. An equilateral triangle is drawn so that no point of the triangle lies outside . The maximum possible area of such a triangle can be written in the form 
, where 
, 
, and 
are positive integers, and is not divisible by the square of any prime number. Find 
.
Solution
Consider points on the complex plane 
. Since the rectangle is quite close to a square, we figure that the area of the equilateral triangle is maximized when a vertex of the triangle coincides with that of the rectangle. Set one vertex of the triangle at 
, and the other two points 
and 
on 
and 
, respectively. Let 
and 
. Since it's equilateral, then 
, so 
, and expanding we get 
.
We can then set the real and imaginary parts equal, and solve for 
. Hence a side 
of the equilateral triangle can be found by 
. Using the area formula 
, the area of the equilateral triangle is 
. Thus 
.
Solution
alternatively
Let angle EAB =x . then angle DFA=60+x. Let the side of the equilateral triangle be s so s= (a^2+121)^ .5
therefore sinx=a/s and sin(60+x)=10/s... therfore 10/s =sin(60+x)=sin60cosx+cos60sinx =(3^.5/2)(11/s)+(1/2)(a/s) multiplying both sides by 2s we get 20=11(3^.5)+a so a=20-11(3^.5)
sustituting this into area=(a^2+121)(3^.5/4) we get the desired answer
See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Last Question | |
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