2006 iTest Problems/Problem 27

Problem

Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\sqrt n$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Determine $m+n$ .

$[asy] import olympiad; size(250); defaultpen(linewidth(0.7)+fontsize(11pt)); real r = 31, t = -10; pair A = origin, B = dir(r-60), C = dir(r); pair X = -0.8 * dir(t), Y = 2 * dir(t); pair D = foot(B,X,Y), E = foot(C,X,Y); draw(A--B--C--A^^X--Y^^B--D^^C--E); label("$A$",A,S); label("$B$",B,S); label("$C$",C,N); label("$D$",D,dir(B--D)); label("$E$",E,dir(C--E)); [/asy]$

Solution

Let $X$ be the intercept of $CB$ and $ED$ . By the Vertical Angle Theorem, $\angle CXE = \angle DXB$ . Also, since both $CE$ and $BD$ are perpendicular to $AD$ , $CE \parallel DB$ . Thus, $\triangle CEX \sim \triangle BDX$ by AA Similarity. Since $2BD = CE$ , $DX = \tfrac13$ and $EX = \tfrac23$ .

Let $a$ be the side length of the triangle, so $BX = \tfrac{a}{3}$ . By the Pythagorean Theorem, $DB = \tfrac{\sqrt{a^2 - 1}}{3}$ . Also, $\angle ABC = 60^\circ$ , so by the Law of Cosines, $AX = \sqrt{a^2 + \tfrac19 a^2 - \tfrac23 a^2 \cdot \tfrac12} = \tfrac{a\sqrt{7}}{3}$ .

$[asy] pair A=(0,0), X=(14/3,0), D=(5,0), B=(5,-1.732); draw(A--D--B--A); dot(A); label("A",A,NW); dot(D); label("D",D,NE); dot(B); label("B",B,SE); dot(X); label("X",X,NW); label("$\frac{a\sqrt{7}}{3}$",(7/3,0),N); label("$\frac13$",(14.5/3,0),N); label("$\frac{\sqrt{a^2-1}}{3}$",(5,-0.866),E); label("$a$",(2.5,-0.866), SW); [/asy]$

By using the Pythagorean Theorem again, we have $\begin{align*} \left( \frac{a\sqrt{7} + 1}{3} \right)^2 + \left( \frac{\sqrt{a^2 - 1}}{3} \right)^2 &= a^2 \\ (a\sqrt{7} + 1)^2 + a^2 - 1 &= 9a^2 \\ 7a^2 + 2a\sqrt{7} + 1 + a^2 - 1 &= 9a^2 \\ 2a\sqrt{7} &= a^2 \\ a &= 2\sqrt{7} \end{align*}$ Thus, the area of the triangle is $\tfrac{28\sqrt{3}}{4} = 7\sqrt{3}$ , so $m+n = \boxed{10}$ .

2006 iTest (Problems, Answer Key)
Preceded by: Problem 26	Followed by: Problem 28
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2006 iTest Problems/Problem 27

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