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Difference between revisions of "2007 AIME I Problems/Problem 15"

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Denote the length of a side of the triangle <math>x</math>, and of <math>\overline{AE}</math> as <math>y</math>. The area of the entire equilateral triangle is <math>\frac{x^2\sqrt{3}}{4}</math>. Add up the areas of the triangles using the <math>\frac{1}{2}ab\sin C</math> formula (notice that for the three outside triangles, <math>\sin 60 = \frac{\sqrt{3}}{2}</math>): <math>\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5 \cdot y + (x - 2)(x - 5) + 2(x - y)) + 14\sqrt{3}</math>. This simplifies to <math>\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5y + x^2 - 7x + 10 + 2x - 2y + 56)</math>. Some terms will cancel out, leaving <math>y = \frac{5}{3}x - 22</math>.
 
Denote the length of a side of the triangle <math>x</math>, and of <math>\overline{AE}</math> as <math>y</math>. The area of the entire equilateral triangle is <math>\frac{x^2\sqrt{3}}{4}</math>. Add up the areas of the triangles using the <math>\frac{1}{2}ab\sin C</math> formula (notice that for the three outside triangles, <math>\sin 60 = \frac{\sqrt{3}}{2}</math>): <math>\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5 \cdot y + (x - 2)(x - 5) + 2(x - y)) + 14\sqrt{3}</math>. This simplifies to <math>\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5y + x^2 - 7x + 10 + 2x - 2y + 56)</math>. Some terms will cancel out, leaving <math>y = \frac{5}{3}x - 22</math>.
  
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<math>\angle FEC</math> is an [[external angle]] to <math>\triangle AEF</math>, from which we find that <math>\displaystyle 60 + \angle CED = 60 + \angle AFE</math>, so <math>\displaystyle \angle CED = \angle AFE</math>. Similarly, we find that <math>\angle EDC = \angle AEF</math>. Thus, <math>\triangle AEF \sim \triangle CDE</math>. Setting up a [[ratio]] of sides, we get that <math>\frac{5}{x-y} = \frac{y}{2}</math>. Using the previous relationship between <math>x</math> and <math>y</math>, we can solve for <math>x</math>.
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<math>\angle FEC</math> is an [[exterior angle]] to <math>\triangle AEF</math>, from which we find that <math>60 + \angle CED = 60 + \angle AFE</math>, so <math>\angle CED = \angle AFE</math>. Similarly, we find that <math>\angle EDC = \angle AEF</math>. Thus, <math>\triangle AEF \sim \triangle CDE</math>. Setting up a [[ratio]] of sides, we get that <math>\frac{5}{x-y} = \frac{y}{2}</math>. Using the previous relationship between <math>x</math> and <math>y</math>, we can solve for <math>x</math>.
 
   
 
   
 
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<div style="text-align:center;">
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<math>\displaystyle xy - y^2 = 10</math>
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<math>xy - y^2 = 10</math>
  
 
<math>\frac{5}{3}x^2 - 22x - \left(\frac{5}{3}x - 22\right)^2 - 10 = 0</math>
 
<math>\frac{5}{3}x^2 - 22x - \left(\frac{5}{3}x - 22\right)^2 - 10 = 0</math>
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</div>
 
</div>
  
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Use the [[quadratic formula]], though we only need the root of the [[discriminant]]. This is <math>\sqrt{(7 \cdot 66)^2 - 4 \cdot 10 \cdot (66^2 + 90)} = \sqrt{49 \cdot 66^2 - 40 \cdot 66^2 - 4 \cdot 9 \cdot 100}</math><math> = \sqrt{9 \cdot 4 \cdot 33^2 - 9 \cdot 4 \cdot 100} = 6\sqrt{33^2 - 100}</math>. The answer is <math>989</math>.
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Use the [[quadratic formula]], though we only need the root of the [[discriminant]]. This is <math>\sqrt{(7 \cdot 66)^2 - 4 \cdot 10 \cdot (66^2 + 90)} = \sqrt{49 \cdot 66^2 - 40 \cdot 66^2 - 4 \cdot 9 \cdot 100}</math><math> = \sqrt{9 \cdot 4 \cdot 33^2 - 9 \cdot 4 \cdot 100} = 6\sqrt{33^2 - 100}</math>. The answer is <math>\boxed{989}</math>.
 +
 
 +
==Solution 2==
 +
First of all, assume <math>EC=x,BD=m, ED=a, EF=b</math>, then we can find <math>BF=m-3, AE=2+m-x</math>
 +
It is not hard to find <math>ab*sin60^{\circ}*\frac{1}{2}=14\sqrt{3}, ab=56</math>, we apply LOC on <math>\triangle{DEF}, \triangle{BFD}</math>, getting that <math>(m-3)^2+m^2-m(m-3)=a^2+b^2-ab</math>, leads to <math>a^2+b^2=m^2-3m+65</math>
 +
Apply LOC on <math>\triangle{CED}, \triangle{AEF}</math> separately, getting <math>4+x^2-2x=a^2; 25+(2+m-x)^2-5(2+m-x)=b^2.</math> Add those terms together and use the equality <math>a^2+b^2=m^2-3m+65</math>, we can find:
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<math>2x^2-(2m+1)x+2m-42=0</math>
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 +
According to basic angle chasing, <math>\angle{A}=\angle{C}; \angle{AFE}=\angle{CED}</math>, so <math>\triangle{AFE}\sim \triangle{CED}</math>, the ratio makes <math>\frac{5}{x}=\frac{2+m-x}{2}</math>, getting that <math>x^2-(2+m)x+10=0</math>
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Now we have two equations with <math>m</math>, and <math>x</math> values for both equations must be the same, so we can solve for <math>x</math> in two equations.
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<math>x=\frac{2m+1 \pm \sqrt{4m^2+4m+1-16m+336}}{4}; x=\frac{4+2m \pm \sqrt{4m^2+16m-144}}{4}</math>, then we can just use positive sign to solve, simplifies to <math>3+\sqrt{4m^2+16m-144}=\sqrt{4m^2-12m+337}</math>, getting <math>m=\frac{211-3\sqrt{989}}{10}</math>, since the triangle is equilateral, <math>AB=BC=2+m=\frac{231-3\sqrt{989}}{10}</math>, and the desired answer is <math>\boxed{989}</math>
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 +
~bluesoul
  
 
== See also ==
 
== See also ==
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[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 21:14, 6 August 2022

Problem

Let $ABC$ be an equilateral triangle, and let $D$ and $F$ be points on sides $BC$ and $AB$, respectively, with $FA = 5$ and $CD = 2$. Point $E$ lies on side $CA$ such that angle $DEF = 60^{\circ}$. The area of triangle $DEF$ is $14\sqrt{3}$. The two possible values of the length of side $AB$ are $p \pm q \sqrt{r}$, where $p$ and $q$ are rational, and $r$ is an integer not divisible by the square of a prime. Find $r$.

Solution

AIME I 2007-15.png

Denote the length of a side of the triangle $x$, and of $\overline{AE}$ as $y$. The area of the entire equilateral triangle is $\frac{x^2\sqrt{3}}{4}$. Add up the areas of the triangles using the $\frac{1}{2}ab\sin C$ formula (notice that for the three outside triangles, $\sin 60 = \frac{\sqrt{3}}{2}$): $\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5 \cdot y + (x - 2)(x - 5) + 2(x - y)) + 14\sqrt{3}$. This simplifies to $\frac{x^2\sqrt{3}}{4} = \frac{\sqrt{3}}{4}(5y + x^2 - 7x + 10 + 2x - 2y + 56)$. Some terms will cancel out, leaving $y = \frac{5}{3}x - 22$.

$\angle FEC$ is an exterior angle to $\triangle AEF$, from which we find that $60 + \angle CED = 60 + \angle AFE$, so $\angle CED = \angle AFE$. Similarly, we find that $\angle EDC = \angle AEF$. Thus, $\triangle AEF \sim \triangle CDE$. Setting up a ratio of sides, we get that $\frac{5}{x-y} = \frac{y}{2}$. Using the previous relationship between $x$ and $y$, we can solve for $x$.

$xy - y^2 = 10$

$\frac{5}{3}x^2 - 22x - \left(\frac{5}{3}x - 22\right)^2 - 10 = 0$

$\frac{5}{3}x^2 - \frac{25}{9}x^2 - 22x + 2 \cdot \frac{5 \cdot 22}{3}x - 22^2 - 10= 0$

$10x^2 - 462x + 66^2 + 90 = 0$

Use the quadratic formula, though we only need the root of the discriminant. This is $\sqrt{(7 \cdot 66)^2 - 4 \cdot 10 \cdot (66^2 + 90)} = \sqrt{49 \cdot 66^2 - 40 \cdot 66^2 - 4 \cdot 9 \cdot 100}$$= \sqrt{9 \cdot 4 \cdot 33^2 - 9 \cdot 4 \cdot 100} = 6\sqrt{33^2 - 100}$. The answer is $\boxed{989}$.

Solution 2

First of all, assume $EC=x,BD=m, ED=a, EF=b$, then we can find $BF=m-3, AE=2+m-x$ It is not hard to find $ab*sin60^{\circ}*\frac{1}{2}=14\sqrt{3}, ab=56$, we apply LOC on $\triangle{DEF}, \triangle{BFD}$, getting that $(m-3)^2+m^2-m(m-3)=a^2+b^2-ab$, leads to $a^2+b^2=m^2-3m+65$ Apply LOC on $\triangle{CED}, \triangle{AEF}$ separately, getting $4+x^2-2x=a^2; 25+(2+m-x)^2-5(2+m-x)=b^2.$ Add those terms together and use the equality $a^2+b^2=m^2-3m+65$, we can find: $2x^2-(2m+1)x+2m-42=0$

According to basic angle chasing, $\angle{A}=\angle{C}; \angle{AFE}=\angle{CED}$, so $\triangle{AFE}\sim \triangle{CED}$, the ratio makes $\frac{5}{x}=\frac{2+m-x}{2}$, getting that $x^2-(2+m)x+10=0$ Now we have two equations with $m$, and $x$ values for both equations must be the same, so we can solve for $x$ in two equations. $x=\frac{2m+1 \pm \sqrt{4m^2+4m+1-16m+336}}{4}; x=\frac{4+2m \pm \sqrt{4m^2+16m-144}}{4}$, then we can just use positive sign to solve, simplifies to $3+\sqrt{4m^2+16m-144}=\sqrt{4m^2-12m+337}$, getting $m=\frac{211-3\sqrt{989}}{10}$, since the triangle is equilateral, $AB=BC=2+m=\frac{231-3\sqrt{989}}{10}$, and the desired answer is $\boxed{989}$

~bluesoul

See also

2007 AIME I (Problems • Answer Key • Resources)
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