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Difference between revisions of "2013 AIME I Problems/Problem 9"
(→‎Solution 3 (Coordinate Bash))
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== Solution 3 (Coordinate Bash) ==
 
== Solution 3 (Coordinate Bash) ==
−
See here: http://artofproblemsolving.com/community/c24001h1060659_2013_aime_i_problem_9
+
 
 +
e let the original position of <math>A</math> be <math>A</math>, and the position of <math>A</math> after folding be <math>D</math>. Also, we put the triangle on the coordinate plane such that <math>A=(0,0)</math>, <math>B=(-6,-6\sqrt3)</math>, <math>C=(6,-6\sqrt3)</math>, and <math>D=(3,-6\sqrt3)</math>.
 +
 
 +
[asy]
 +
import cse5;
 +
size(10cm);
 +
pen tpen = defaultpen + 1.337;
 +
real a = 39/5.0;
 +
real b = 39/7.0;
 +
pair B = MP("B", (0,0), dir(200));
 +
pair A = (9,0);
 +
pair C = MP("C", (12,0), dir(-20));
 +
pair K = (6,10.392);
 +
pair M = (a*B+(12-a)*K) / 12;
 +
pair N = (b*C+(12-b)*K) / 12;
 +
draw(B--M--N--C--cycle);
 +
draw(M--A--N--cycle);
 +
label("<math>D</math>", A, S);
 +
pair X = (6,6*sqrt(3));
 +
draw(B--X--C);
 +
label("<math>A</math>",X,dir(90));
 +
draw(A--X);
 +
[/asy]
 +
 
 +
Note that since <math>A</math> is reflected over the fold line to <math>D</math>, the fold line is the perpendicular bisector of <math>AD</math>. We know <math>A=(0,0)</math> and <math>D=(3,-6\sqrt3)</math>. The midpoint of <math>AD</math> (which is a point on the fold line) is <math>(\tfrac32, -3\sqrt3)</math>. Also, the slope of <math>AD</math> is <math>\frac{-6\sqrt3}{3}=-2\sqrt3</math>, so the slope of the fold line (which is perpendicular), is the negative of the reciprocal of the slope of <math>AD</math>, or <math>\frac{1}{2\sqrt3}=\frac{\sqrt3}{6}</math>. Then, using point slope form, the equation of the fold line is
 +
<cmath>y+3\sqrt3=\frac{\sqrt3}{6}\left(x-\frac32\right)</cmath><cmath>y=\frac{\sqrt3}{6}x-\frac{13\sqrt3}{4}</cmath>
 +
Note that the equations of lines <math>AB</math> and <math>AC</math> are <math>y=\sqrt3x</math> and <math>y=-\sqrt3x</math>, respectively. We will first find the intersection of <math>AB</math> and the fold line by substituting for <math>y</math>:
 +
<cmath>\sqrt3 x=\frac{\sqrt3}{6}x-\frac{13\sqrt3}{4}</cmath><cmath>\frac{5\sqrt3}{6}x=-\frac{13\sqrt3}{4} \implies x=-\frac{39}{10}</cmath>
 +
Therefore, the point of intersection is <math>\left(-\tfrac{39}{10},-\tfrac{39\sqrt3}{10}\right)</math>. Now, lets find the intersection with <math>AC</math>. Substituting for <math>y</math> yields
 +
<cmath>-\sqrt3 x=\frac{\sqrt3}{6}x-\frac{13\sqrt3}{4}</cmath><cmath>\frac{-7\sqrt3}{6}x=-\frac{13\sqrt3}{4} \implies x=\frac{39}{14}</cmath>
 +
Therefore, the point of intersection is <math>\left(\tfrac{39}{14},-\tfrac{39\sqrt3}{14}\right)</math>. Now, we just need to use the distance formula to find the distance between <math>\left(-\tfrac{39}{10},-\tfrac{39\sqrt3}{10}\right)</math> and <math>\left(\tfrac{39}{14},-\tfrac{39\sqrt3}{14}\right)</math>.
 +
<cmath>\sqrt{\left(\frac{39}{14}+\frac{39}{10}\right)^2+\left(-\frac{39\sqrt3}{14}+\frac{39\sqrt3}{10}\right)^2}</cmath>
 +
The number 39 is in all of the terms, so let's factor it out:
 +
<cmath>39\sqrt{\left(\frac{1}{14}+\frac{1}{10}\right)^2+\left(-\frac{\sqrt3}{14}+\frac{\sqrt3}{10}\right)^2}=39\sqrt{\left(\frac{6}{35}\right)^2+\left(\frac{\sqrt3}{35}\right)^2}</cmath><cmath>\frac{39}{35}\sqrt{6^2+\sqrt3^2}=\frac{39\sqrt{39}}{35}</cmath>
 +
Therefore, our answer is <math>39+39+35=\boxed{113}</math>, and we are done.
 +
 
 +
Solution by nosaj.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2013|n=I|num-b=8|num-a=10}}
 
{{AIME box|year=2013|n=I|num-b=8|num-a=10}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 14:52, 26 May 2017

Problem 9

A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.

[asy] import cse5; size(12cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = MP("A", (9,0), dir(-80)); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle, tpen); draw(M--A--N--cycle); fill(M--A--N--cycle, mediumgrey); pair shift = (-20.13, 0); pair B1 = MP("B", B+shift, dir(200)); pair A1 = MP("A", K+shift, dir(90)); pair C1 = MP("C", C+shift, dir(-20)); draw(A1--B1--C1--cycle, tpen);[/asy]

Solution 1

Let $P$ and $Q$ be the points on $\overline{AB}$ and $\overline{AC}$, respectively, where the paper is folded.

Let $D$ be the point on $\overline{BC}$ where the folded $A$ touches it.

Let $a$, $b$, and $x$ be the lengths $AP$, $AQ$, and $PQ$, respectively.

We have $PD = a$, $QD = b$, $BP = 12 - a$, $CQ = 12 - b$, $BD = 9$, and $CD = 3$.

Using the Law of Cosines on $BPD$:

$a^{2} = (12 - a)^{2} + 9^{2} - 2 \times (12 - a) \times 9 \times \cos{60}$

$a^{2} = 144 - 24a + a^{2} + 81 - 108 + 9a$

$a = \frac{39}{5}$

Using the Law of Cosines on $CQD$:

$b^{2} = (12 - b)^{2} +3^{2} - 2 \times (12 - b) \times 3 \times \cos{60}$

$b^{2} = 144 - 24b + b^{2} + 9 - 36 + 3b$

$b = \frac{39}{7}$

Using the Law of Cosines on $APQ$:

$x^{2} = a^{2} + b^{2} - 2ab \cos{60}$

$x^{2} = (\frac{39}{5})^2 + (\frac{39}{7})^2 - (\frac{39}{5} \times \frac{39}{7})$

$x = \frac{39 \sqrt{39}}{35}$

The solution is $39 + 39 + 35 = \boxed{113}$.

Solution 2

Proceed with the same labeling as in Solution 1.

$\angle B = \angle C = \angle A = \angle PDQ = 60^\circ$

$\angle PDB + \angle PDQ + \angle QDC = \angle QDC + \angle CQD + \angle C = 180^\circ$

Therefore, $\angle PDB = \angle CQD$.

Similarly, $\angle BPD = \angle QDC$.

Now, $\bigtriangleup BPD$ and $\bigtriangleup CDQ$ are similar triangles, so

$\frac{3}{12-a} = \frac{12-b}{9} = \frac{b}{a}$.

Solving this system of equations yields $a = \frac{39}{5}$ and $b = \frac{39}{7}$.

Using the Law of Cosines on $APQ$:

$x^{2} = a^{2} + b^{2} - 2ab \cos{60}$

$x^{2} = (\frac{39}{5})^2 + (\frac{39}{7})^2 - (\frac{39}{5} \times \frac{39}{7})$

$x = \frac{39 \sqrt{39}}{35}$

The solution is $39 + 39 + 35 = \boxed{113}$.

Solution 3 (Coordinate Bash)

e let the original position of $A$ be $A$, and the position of $A$ after folding be $D$. Also, we put the triangle on the coordinate plane such that $A=(0,0)$, $B=(-6,-6\sqrt3)$, $C=(6,-6\sqrt3)$, and $D=(3,-6\sqrt3)$.

[asy] import cse5; size(10cm); pen tpen = defaultpen + 1.337; real a = 39/5.0; real b = 39/7.0; pair B = MP("B", (0,0), dir(200)); pair A = (9,0); pair C = MP("C", (12,0), dir(-20)); pair K = (6,10.392); pair M = (a*B+(12-a)*K) / 12; pair N = (b*C+(12-b)*K) / 12; draw(B--M--N--C--cycle); draw(M--A--N--cycle); label("$D$", A, S); pair X = (6,6*sqrt(3)); draw(B--X--C); label("$A$",X,dir(90)); draw(A--X); [/asy]

Note that since $A$ is reflected over the fold line to $D$, the fold line is the perpendicular bisector of $AD$. We know $A=(0,0)$ and $D=(3,-6\sqrt3)$. The midpoint of $AD$ (which is a point on the fold line) is $(\tfrac32, -3\sqrt3)$. Also, the slope of $AD$ is $\frac{-6\sqrt3}{3}=-2\sqrt3$, so the slope of the fold line (which is perpendicular), is the negative of the reciprocal of the slope of $AD$, or $\frac{1}{2\sqrt3}=\frac{\sqrt3}{6}$. Then, using point slope form, the equation of the fold line is \[y+3\sqrt3=\frac{\sqrt3}{6}\left(x-\frac32\right)\]\[y=\frac{\sqrt3}{6}x-\frac{13\sqrt3}{4}\] Note that the equations of lines $AB$ and $AC$ are $y=\sqrt3x$ and $y=-\sqrt3x$, respectively. We will first find the intersection of $AB$ and the fold line by substituting for $y$: \[\sqrt3 x=\frac{\sqrt3}{6}x-\frac{13\sqrt3}{4}\]\[\frac{5\sqrt3}{6}x=-\frac{13\sqrt3}{4} \implies x=-\frac{39}{10}\] Therefore, the point of intersection is $\left(-\tfrac{39}{10},-\tfrac{39\sqrt3}{10}\right)$. Now, lets find the intersection with $AC$. Substituting for $y$ yields \[-\sqrt3 x=\frac{\sqrt3}{6}x-\frac{13\sqrt3}{4}\]\[\frac{-7\sqrt3}{6}x=-\frac{13\sqrt3}{4} \implies x=\frac{39}{14}\] Therefore, the point of intersection is $\left(\tfrac{39}{14},-\tfrac{39\sqrt3}{14}\right)$. Now, we just need to use the distance formula to find the distance between $\left(-\tfrac{39}{10},-\tfrac{39\sqrt3}{10}\right)$ and $\left(\tfrac{39}{14},-\tfrac{39\sqrt3}{14}\right)$. \[\sqrt{\left(\frac{39}{14}+\frac{39}{10}\right)^2+\left(-\frac{39\sqrt3}{14}+\frac{39\sqrt3}{10}\right)^2}\] The number 39 is in all of the terms, so let's factor it out: \[39\sqrt{\left(\frac{1}{14}+\frac{1}{10}\right)^2+\left(-\frac{\sqrt3}{14}+\frac{\sqrt3}{10}\right)^2}=39\sqrt{\left(\frac{6}{35}\right)^2+\left(\frac{\sqrt3}{35}\right)^2}\]\[\frac{39}{35}\sqrt{6^2+\sqrt3^2}=\frac{39\sqrt{39}}{35}\] Therefore, our answer is $39+39+35=\boxed{113}$, and we are done.

Solution by nosaj.

See also

2013 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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All AIME Problems and Solutions

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