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Difference between revisions of "1997 AIME Problems/Problem 15"

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[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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alternatively
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Let angle EAB =x . then angle DFA=60+x.  Let the side of the equilateral triangle be s so s= (a^2+121)^ .5
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therefore sinx=a/s  and sin(60+x)=10/s...
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therfore 10/s =sin(60+x)=sin60cosx+cos60sinx =(3^.5/2)(11/s)+(1/2)(a/s)
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multiplying both sides by 2s we get 20=11(3^.5)+a  so a=20-11(3^.5)
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sustituting this into area=(a^2+121)(3^.5/4)  we get the desired answer

Revision as of 16:09, 21 July 2008

Problem

The sides of rectangle $ABCD$ have lengths $10$ and $11$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. The maximum possible area of such a triangle can be written in the form $p\sqrt{q}-r$, where $p$, $q$, and $r$ are positive integers, and $q$ is not divisible by the square of any prime number. Find $p+q+r$.

Solution

Consider points on the complex plane $A (0,0),\ B (11,0),\ C (11,10),\ D (0,10)$. 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 $A$, and the other two points $E$ and $F$ on $BC$ and $CD$, respectively. Let $E (11,a)$ and $F (b, 10)$. Since it's equilateral, then $E\cdot\text{cis}60^{\circ} = F$, so $(11 + ai)\left(\frac {1}{2} + \frac {\sqrt {3}}{2}i\right) = b + 10i$, and expanding we get $\left(\frac {11}{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i$.

1997 AIME-15a.PNG

We can then set the real and imaginary parts equal, and solve for $(a,b) = (20 - 11\sqrt {3},22 - 10\sqrt {3})$. Hence a side $s$ of the equilateral triangle can be found by $s^2 = AE^2 = a^2 + AB^2 = 884 - 440\sqrt{3}$. Using the area formula $\frac{s^2\sqrt{3}}{4}$, the area of the equilateral triangle is $\frac{(884-440\sqrt{3})\sqrt{3}}{4} = 221\sqrt{3} - 330$. Thus $p + q + r = 221 + 3 + 330 = \boxed{554}$.

See also

1997 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Question
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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

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