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Mock AIME II 2012 Problems/Problem 9

Problem

In $\triangle ABC$, $AB=12$, $AC=20$, and $\angle ABC=120^\circ$. $D, E,$ and $F$ lie on $\overline{AC}, \overline{AB}$, and $\overline{BC}$, respectively. If $AE=\frac{1}{4}AB, BF=\frac{1}{4}BC$, and $AD=\frac{1}{4}AC$, the area of $\triangle DEF$ can be expressed in the form $\frac{a\sqrt{b}-c\sqrt{d}}{e}$ where $a, b, c, d, e$ are all positive integers, and $b$ and $d$ do not have any perfect squares greater than $1$ as divisors. Find $a+b+c+d+e$.

Solution

Here is a diagram (note that D should be on AC and F should be on BC): [asy] /* File unicodetex not found. */ /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(12cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -24, xmax = 24, ymin = -15, ymax = 22; /* image dimensions */ pen qqwuqq = rgb(0,0,0); draw(arc((12,0),2,60,180)--(12,0)--cycle, qqwuqq); /* draw figures */ draw((0,0)--(12,0)); draw((0,0)--(18,10)); draw((18,10)--(12,0)); label("12",(6,-1),SE*labelscalefactor); label("20",(8,7),SE*labelscalefactor); draw((5,3)--(14,3)); draw((5,3)--(3,0)); draw((3,0)--(14,3)); /* dots and labels */ dot((0,0),dotstyle); label("$A$", (0,0), NE * labelscalefactor); dot((12,0),dotstyle); label("$B$", (12,0), NE * labelscalefactor); dot((18,10),dotstyle); label("$C$", (18,10), NE * labelscalefactor); label("$120^\circ$", (11,1), NE * labelscalefactor,qqwuqq); dot((3,0),dotstyle); label("$E$", (3,0), NE * labelscalefactor); dot((14,3),dotstyle); label("$F$", (14,3), NE * labelscalefactor); dot((5,3),dotstyle); label("$D$", (5,3), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] . Start out by finding $BC$. Remark that by the law of sines, $\frac{\sin(120^\circ)}{20}=\frac{\sin(\angle ACB)}{12}$. Therefore $\sin(\angle ACB)=\frac{3}{5}*\frac{\sqrt{3}}{2}=\frac{3\sqrt3}{10}$. We know $90^\circ>\angle ACB>0^\circ$ because $\angle ABC>90^\circ$ in $\triangle ABC$, therefore $\angle ACB$ is in the first quadrant. Use the Pythagorean Identity to give us $\cos^2(\angle ACB)+\sin^2(\angle ACB)=1\implies \cos(\angle ACB)=\frac{\sqrt{73}}{10}$.

Now, note that $\angle BAC$ is the same thing as $60^\circ-\angle ACB$, from $\triangle ABC$, therefore we have $\sin(\angle BAC)=\sin(60^\circ-\angle ACB)=\sin(60^\circ)\cos(\angle ACB)-\sin(\angle ACB)\cos(60^\circ)$ $\equal{} \frac{\sqrt3*\sqrt{73}}{20}-\frac{3\sqrt3}{20}=\frac{\sqrt3*\sqrt{73}-3\sqrt3}{20}$ (Error compiling LaTeX. Unknown error_msg).

Next, use the law of sines to give us $\frac{\sin(120^\circ)}{20}=\frac{\sin(\angle BAC)}{BC}$. This gives us $\frac{\sqrt3}{40}=\frac{\sqrt3*\sqrt{73}-3\sqrt3}{20BC}$. This gives us $2\sqrt{73}-6=BC$.

Now, we use coordinates to find that the coordinates of $C$, $E$, $F$, and $D$. $C$ is going to be the point $(9+\sqrt{73}, \sqrt{219}-3\sqrt3)$ from adding point $H$ to create $30^\circ, 60^\circ, 90^\circ$ triangle $BCH$ as shown in this diagram (where $H$ is the right angle, $B$ is the $30^\circ$ angle, and $C$ is the $60^\circ$ angle):

[asy] import graph; usepackage("amsmath"); size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -24, xmax = 24, ymin = -15, ymax = 22; /* image dimensions */ pen qqwuqq = rgb(0,0,0); draw(arc((12,0),2,60,180)--(12,0)--cycle, qqwuqq); /* draw figures */ draw((0,0)--(12,0)); draw((0,0)--(18,10)); draw((18,10)--(12,0)); label("12",(6,-1),SE*labelscalefactor); label("$ 2\sqrt{76}-6 $",(14,5),SE*labelscalefactor); label("20",(8,7),SE*labelscalefactor); draw((12,0)--(12,10)); draw((12,10)--(18,10)); /* dots and labels */ dot((0,0),dotstyle); label("$A$", (0,0), NE * labelscalefactor); dot((12,0),dotstyle); label("$B$", (12,0), NE * labelscalefactor); dot((18,10),dotstyle); label("$C$", (18,10), NE * labelscalefactor); label("$120^\circ$", (11,1), NE * labelscalefactor,qqwuqq); dot((12,10),dotstyle); label("$H$", (12,10), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Now, note that we have to find the coordinates of $E,F$ and $D$. Assume WLOG that $A$ is $(0,0)$ and $B$ be $(12,0)$. $E$ is obviously $(3,0)$, $F$ is going to be $(12+\frac{9+\sqrt{73}-12}{4}, \frac{\sqrt{219}-3\sqrt3}{4})$ and the coordinates of $D$ is going to be $(\frac{9+\sqrt{73}}{4}, \frac{\sqrt{219}-3\sqrt3}{4})$. Since $D$ and $F$ have the same $y$ value, we can find $[EDF]$ by using $\frac12*b*h$. The base is going to be equal to the distance from $D$ to $F$, which is the same thing as $12+\frac{9+\sqrt{73}-12}{4}-\frac{9+\sqrt{73}}{4}=9=b$, and the height is the change in the $y$ coordinate from $E$ to $D$. This is the same thing as $\frac{\sqrt{219}-3\sqrt3}{4}-0=\frac{\sqrt{219}-3\sqrt3}{4}=h$. Hence, plugging these into $[EDF]=\frac12*b*h$ give us $\frac12*9*\frac{\sqrt{219}-3\sqrt3}{4}=\frac{9\sqrt{219}-27\sqrt3}{8}$. The answer is thus $9+219+27+3+8=\boxed{266}$.

Solution 2

Let $S$ be the area of the large triangle and let $x$ be the area we are trying to find. We have that $\Delta ADE$ is similar to $\Delta ABC$ with a ratio of $1:4$ while $\Delta CDF$ is similar to $\Delta ABC$ with a ratio of $3:4$. Then the area of $\Delta ADE = \frac{S}{16}$ and the area of $\Delta CDE = \frac{9S}{16}$. Lastly, the area of $\Delta EBF = \frac{1}{2}EB*BF*\sin B = \frac{1}{2}* \frac{3AB}{4}*\frac{BC}{4}*\sin B = \frac{3}{16}S$. The area of $\Delta ABC$ is equal to the sum of the areas of the four smaller triangles so $S= \frac{S}{16}+\frac{9S}{16}+\frac{3S}{16} + x$. Thus $x = \frac{3S}{16}$.

It remains to find $S$. First we can use the Law of Cosines on $\Delta ABC$ to find $12^2+BC^2 + 12BC = 400$ which solves to $BC = -6 + 2\sqrt{73}$. Then $S = \frac{1}{2} AB* BC*\sin{120} = -18\sqrt3 + 6\sqrt{219}$. Then substituting yields $x =\frac{-27\sqrt 3 + 9 \sqrt{219}}{8}$ so the answer is $\boxed{266}$

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