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Difference between revisions of "2013 AIME II Problems/Problem 4"

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==Problem 4==
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==Problem==
 
In the Cartesian plane let <math>A = (1,0)</math> and <math>B = \left( 2, 2\sqrt{3} \right)</math>.  Equilateral triangle <math>ABC</math> is constructed so that <math>C</math> lies in the first quadrant.  Let <math>P=(x,y)</math> be the center of <math>\triangle ABC</math>.  Then <math>x \cdot y</math> can be written as <math>\tfrac{p\sqrt{q}}{r}</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is an integer that is not divisible by the square of any prime.  Find <math>p+q+r</math>.
 
In the Cartesian plane let <math>A = (1,0)</math> and <math>B = \left( 2, 2\sqrt{3} \right)</math>.  Equilateral triangle <math>ABC</math> is constructed so that <math>C</math> lies in the first quadrant.  Let <math>P=(x,y)</math> be the center of <math>\triangle ABC</math>.  Then <math>x \cdot y</math> can be written as <math>\tfrac{p\sqrt{q}}{r}</math>, where <math>p</math> and <math>r</math> are relatively prime positive integers and <math>q</math> is an integer that is not divisible by the square of any prime.  Find <math>p+q+r</math>.
  
 
==Solution 1 ==
 
==Solution 1 ==
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The distance from point <math>A</math> to point <math>B</math> is <math> \sqrt{13}</math>. The vector that starts at point A and ends at point B is given by <math>B - A = (1, 2\sqrt{3})</math>. Since the center of an equilateral triangle, <math>P</math>, is also the intersection of the perpendicular bisectors of the sides of the triangle, we need first find the equation for the perpendicular bisector to <math>\overline{AB}</math>. The line perpendicular to <math>\overline{AB}</math> through the midpoint, <math>M =  (\dfrac{3}{2},\sqrt{3})</math>, <math>\overline{AB}</math> can be parameterized by <math> (\dfrac{2\sqrt{3}}{\sqrt{13}}, \dfrac{-1}{\sqrt{13}}) t + (\dfrac{3}{2},\sqrt{3})</math>. At this point, it is useful to note that <math>\Delta BMP</math> is a 30-60-90 triangle with <math>\overline{MB}</math> measuring <math>\dfrac{\sqrt{13}}{2}</math>. This yields the length of <math>\overline{MP}</math> to be <math>\dfrac{\sqrt{13}}{2\sqrt{3}}</math>. Therefore, <math>P =( \dfrac{2\sqrt{3}}{\sqrt{13}},\dfrac{-1}{\sqrt{13}})(\dfrac{\sqrt{13}}{2\sqrt{3}}) + (\dfrac{3}{2},\sqrt{3}) = (\dfrac{5}{2}, \dfrac{5}{2\sqrt{3}})</math>. Therefore <math>xy = \dfrac{25\sqrt{3}}{12}</math> yielding an answer of <math> p + q + r  = 25 + 3 + 12 = \boxed{040}</math>.
+
The distance from point <math>A</math> to point <math>B</math> is <math> \sqrt{13}</math>. The vector that starts at point A and ends at point B is given by <math>B - A = (1, 2\sqrt{3})</math>. Since the center of an equilateral triangle, <math>P</math>, is also the intersection of the perpendicular bisectors of the sides of the triangle, we need first find the equation for the perpendicular bisector to <math>\overline{AB}</math>. The line perpendicular to <math>\overline{AB}</math> through the midpoint, <math>M =  \left(\dfrac{3}{2},\sqrt{3}\right)</math>, <math>\overline{AB}</math> can be parameterized by <math>\left(\dfrac{2\sqrt{3}}{\sqrt{13}}, \dfrac{-1}{\sqrt{13}}\right) t + \left(\dfrac{3}{2},\sqrt{3}\right)</math>. At this point, it is useful to note that <math>\Delta BMP</math> is a 30-60-90 triangle with <math>\overline{MB}</math> measuring <math>\dfrac{\sqrt{13}}{2}</math>. This yields the length of <math>\overline{MP}</math> to be <math>\dfrac{\sqrt{13}}{2\sqrt{3}}</math>. Therefore, <math>P =\left(\dfrac{2\sqrt{3}}{\sqrt{13}},\dfrac{-1}{\sqrt{13}}\right)\left(\dfrac{\sqrt{13}}{2\sqrt{3}}\right) + \left(\dfrac{3}{2},\sqrt{3}\right) = \left(\dfrac{5}{2}, \dfrac{5}{2\sqrt{3}}\right)</math>. Therefore <math>xy = \dfrac{25\sqrt{3}}{12}</math> yielding an answer of <math> p + q + r  = 25 + 3 + 12 = \boxed{040}</math>.
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==Solution 2==
 
==Solution 2==
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==Solution 3==
 
==Solution 3==
  
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We can also consider the slopes of the lines. Midpoint <math>M</math> of <math>AB</math> has coordinates <math>\left(\frac{3}{2},\ \sqrt{3}\right)</math>. Because line <math>AB</math> has slope <math>2\sqrt{3}</math>, the slope of line <math>MP</math> is  <math>-\frac{1}{2\sqrt{3}}</math>.  
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We can also consider the slopes of the lines. Midpoint <math>M</math> of <math>AB</math> has coordinates <math>\left(\frac{3}{2},\ \sqrt{3}\right)</math>. Because line <math>AB</math> has slope <math>2\sqrt{3}</math>, the slope of line <math>MP</math> is  <math>-\frac{1}{2\sqrt{3}}</math> (Because of perpendicular slopes).  
  
 
Since <math>\Delta ABC</math> is equilateral, and since point <math>P</math> is the centroid, we can quickly calculate that <math>MP = \frac{\sqrt{39}}{6}</math>. Then, define <math>\Delta x</math> and <math>\Delta y</math> to be the differences between points <math>M</math> and <math>P</math>. Because of the slope, it is clear that <math>\Delta x = 2\sqrt{3} \Delta y</math>.
 
Since <math>\Delta ABC</math> is equilateral, and since point <math>P</math> is the centroid, we can quickly calculate that <math>MP = \frac{\sqrt{39}}{6}</math>. Then, define <math>\Delta x</math> and <math>\Delta y</math> to be the differences between points <math>M</math> and <math>P</math>. Because of the slope, it is clear that <math>\Delta x = 2\sqrt{3} \Delta y</math>.
  
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We can then use the Pythagorean Theorem on line segment <math>MP</math>: <math>MP^2 = \Delta x^2 + \Delta y^2</math> yields <math>\Delta y = -\frac{1}{2\sqrt{3}}</math> and <math>\Delta x = 1</math>, after substituting <math>\Delta x</math>. The coordinates of P are thus <math>\left(\frac{5}{2},\ \frac{5\sqrt{3}}{6}\right)</math>. Multiplying these together gives us <math>\frac{25\sqrt{3}}{12}</math>, giving us <math>040</math> as our answer.
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We can then use the Pythagorean Theorem on line segment <math>MP</math>: <math>MP^2 = \Delta x^2 + \Delta y^2</math> yields <math>\Delta y = -\frac{1}{2\sqrt{3}}</math> and <math>\Delta x = 1</math>, after substituting <math>\Delta x</math>. The coordinates of P are thus <math>\left(\frac{5}{2},\ \frac{5\sqrt{3}}{6}\right)</math>. Multiplying these together gives us <math>\frac{25\sqrt{3}}{12}</math>, giving us <math>\boxed{040}</math> as our answer.
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 +
==Solution 4==
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Since <math>AC</math> will be segment <math>AB</math> rotated clockwise <math>60^{\circ}</math>, we can use a rotation matrix to find <math>C</math>. We first translate the triangle <math>1</math> unit to the left, so <math>A'</math> lies on the origin, and <math>B' = (1, 2\sqrt{3})</math>. Rotating clockwise <math>60^{\circ}</math> is the same as rotating <math>300^{\circ}</math> counter-clockwise, so our rotation matrix is <math>\begin{bmatrix} \cos{300^{\circ}} & -\sin{300^{\circ}}\\ \sin{300^{\circ}} & \cos{300^{\circ}}\\ \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\\ \end{bmatrix}</math>. Then <math>C' = \begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\\ \end{bmatrix} \cdot \begin{bmatrix} 1\\ 2\sqrt{3}\\ \end{bmatrix} = \begin{bmatrix} \frac{7}{2}\\ \frac{\sqrt{3}}{2}\\ \end{bmatrix}</math>. Thus, <math>C = (\frac{9}{2}, \frac{\sqrt{3}}{2})</math>. Since the triangle is equilateral, the center of the triangle is the average of the coordinates of the vertices. Then <math>P = (\frac{1 + 2 + \frac{9}{2}}{3}, \frac{2\sqrt{3} + \frac{\sqrt{3}}{2}}{3}) = (\frac{5}{2}, \frac{5\sqrt{3}}{6})</math>. Our answer is <math>\frac{5}{2} \cdot \frac{5\sqrt{3}}{6} = \frac{25\sqrt{3}}{12}</math>. <math>25 + 3 + 12 = \boxed{40}</math>
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==Solution 5==
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 +
We construct point <math>C</math> by drawing two circles with radius <math>r = AB = \sqrt{13}</math>.  One circle will be centered at <math>A</math>, while the other is centered at <math>B</math>.  The equations of the circles are:
 +
 
 +
<math>(x - 1)^2 + y^2 = 13</math>
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 +
<math>(x - 2)^2 + (y - 2\sqrt{3})^2 = 13</math>
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 +
Setting the LHS of each of these equations equal to each other and solving for <math>x</math> yields after simplification:
 +
 
 +
<math>x = \frac{15}{2} - 2\sqrt{3}y</math>
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 +
Plugging that into the first equation gives the following quadratic in <math>y</math> after simplification:
 +
 
 +
<math>y^2 - 2\sqrt{3}y + \frac{9}{4} = 0</math>
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 +
The quadratic formula gives <math>y = \frac{\sqrt{3}}{2}, \frac{3\sqrt{3}}{2}</math>. 
 +
 
 +
Since <math>x > 0</math> and <math>x = \frac{15}{2} - 2\sqrt{3}y</math>, we pick <math>y = \frac{\sqrt{3}}{2}</math> in the hopes that it will give <math>x > 0</math>.  Plugging <math>y</math> into the equation for <math>x</math> yields <math>x = \frac{9}{2}</math>.
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 +
Thus, <math>C(\frac{9}{2}, \frac{\sqrt{3}}{2})</math>.  Averaging the coordinates of the vertices of equilateral triangle <math>ABC</math> will give the center of mass of the triangle. 
 +
 
 +
Thus, <math>P(\frac{5}{2}, \frac{5\sqrt{3}}{6})</math>, and the product of the coordinates is <math>\frac{25\sqrt{3}}{12}</math>, so the desired quantity is <math>\boxed{040}</math>.
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==Solution 6==
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Labeling our points and sketching a graph we get that <math>C</math> is to the right of <math>AB</math>. Of course, we need to find <math>C</math>. Note that the transformation from <math>A</math> to <math>B</math> is <math>[1,2\sqrt{3}]</math>, and if we imagine a height dropped to <math>AB</math> we see that a transformation from the midpoint <math>(\frac{3}{2},\sqrt {3})</math> to <math>C</math> is basically the first transformation, with <math>\frac{\sqrt{3}}{2}</math> the magnitude and the x and y switched– then multiply the new y by -1. Then, applying this transformation of <math>[3,\frac{-\sqrt{3}}{2}]</math> we get that <math>C=(\frac{9}{2},\frac{\sqrt{3}}{2})</math> which means that <math>P=(\frac{5}{2},\frac{5\sqrt{3}}{6})</math>. Then our answer is <math>\boxed{40}</math>.
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==Solution 7==
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Transform this into the complex plane and let <math>a=1, b=2+2\sqrt3 i</math>. We know that 3 complex numbers <math>a,b,c</math> form an equilateral triangle if <math>a^2+b^2+c^2=ab+bc+ac</math>, so plugging in our values of <math>a,b</math>, we get <math>8\sqrt3 i - 7 +c^2 = 2+2\sqrt3 i + (3+2\sqrt 3i)c.</math> Solving for <math>c</math> using Wolfram Alpha, we find that the solutions are <math>c=\frac 92 + \frac{i\sqrt3}{2}, -\frac 32 + \frac{3i\sqrt3}{2}</math>. The first one is in the first quadrant, so <math>C\left( \frac 92, \frac{\sqrt3}{2} \right)</math>. The center is the average of the coordinates and we find that it is <math>\left(\frac{5}2, \frac{5\sqrt3}{6} \right)</math>. Then <math>xy = \frac{25\sqrt3}{12} \implies 40</math>.
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-bobthegod78, krwang, and Simplest14
  
 
==See Also==
 
==See Also==
 
{{AIME box|year=2013|n=II|num-b=3|num-a=5}}
 
{{AIME box|year=2013|n=II|num-b=3|num-a=5}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:06, 17 October 2024

Problem

In the Cartesian plane let $A = (1,0)$ and $B = \left( 2, 2\sqrt{3} \right)$. Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)$ be the center of $\triangle ABC$. Then $x \cdot y$ can be written as $\tfrac{p\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers and $q$ is an integer that is not divisible by the square of any prime. Find $p+q+r$.

Solution 1

The distance from point $A$ to point $B$ is $\sqrt{13}$. The vector that starts at point A and ends at point B is given by $B - A = (1, 2\sqrt{3})$. Since the center of an equilateral triangle, $P$, is also the intersection of the perpendicular bisectors of the sides of the triangle, we need first find the equation for the perpendicular bisector to $\overline{AB}$. The line perpendicular to $\overline{AB}$ through the midpoint, $M = \left(\dfrac{3}{2},\sqrt{3}\right)$, $\overline{AB}$ can be parameterized by $\left(\dfrac{2\sqrt{3}}{\sqrt{13}}, \dfrac{-1}{\sqrt{13}}\right) t + \left(\dfrac{3}{2},\sqrt{3}\right)$. At this point, it is useful to note that $\Delta BMP$ is a 30-60-90 triangle with $\overline{MB}$ measuring $\dfrac{\sqrt{13}}{2}$. This yields the length of $\overline{MP}$ to be $\dfrac{\sqrt{13}}{2\sqrt{3}}$. Therefore, $P =\left(\dfrac{2\sqrt{3}}{\sqrt{13}},\dfrac{-1}{\sqrt{13}}\right)\left(\dfrac{\sqrt{13}}{2\sqrt{3}}\right) + \left(\dfrac{3}{2},\sqrt{3}\right) = \left(\dfrac{5}{2}, \dfrac{5}{2\sqrt{3}}\right)$. Therefore $xy = \dfrac{25\sqrt{3}}{12}$ yielding an answer of $p + q + r = 25 + 3 + 12 = \boxed{040}$.

Solution 2

Rather than considering the Cartesian plane, we use complex numbers. Thus A is 1 and B is $2 + 2\sqrt{3}i$.

Recall that a rotation of $\theta$ radians counterclockwise is equivalent to multiplying a complex number by $e^{i\theta}$, but here we require a clockwise rotation, so we multiply by $e^{-\frac{i\pi}{3}}$ to obtain C. Upon averaging the coordinates of A, B, and C, we obtain the coordinates of P, viz. $\left(\frac{5}{2}, \frac{5\sqrt{3}}{6}\right)$.

Therefore $xy$ is $\frac{25\sqrt{3}}{12}$ and the answer is $25 + 12 + 3 = \boxed{040}$.

Solution 3

We can also consider the slopes of the lines. Midpoint $M$ of $AB$ has coordinates $\left(\frac{3}{2},\ \sqrt{3}\right)$. Because line $AB$ has slope $2\sqrt{3}$, the slope of line $MP$ is $-\frac{1}{2\sqrt{3}}$ (Because of perpendicular slopes).

Since $\Delta ABC$ is equilateral, and since point $P$ is the centroid, we can quickly calculate that $MP = \frac{\sqrt{39}}{6}$. Then, define $\Delta x$ and $\Delta y$ to be the differences between points $M$ and $P$. Because of the slope, it is clear that $\Delta x = 2\sqrt{3} \Delta y$.

We can then use the Pythagorean Theorem on line segment $MP$: $MP^2 = \Delta x^2 + \Delta y^2$ yields $\Delta y = -\frac{1}{2\sqrt{3}}$ and $\Delta x = 1$, after substituting $\Delta x$. The coordinates of P are thus $\left(\frac{5}{2},\ \frac{5\sqrt{3}}{6}\right)$. Multiplying these together gives us $\frac{25\sqrt{3}}{12}$, giving us $\boxed{040}$ as our answer.

Solution 4

Since $AC$ will be segment $AB$ rotated clockwise $60^{\circ}$, we can use a rotation matrix to find $C$. We first translate the triangle $1$ unit to the left, so $A'$ lies on the origin, and $B' = (1, 2\sqrt{3})$. Rotating clockwise $60^{\circ}$ is the same as rotating $300^{\circ}$ counter-clockwise, so our rotation matrix is $\begin{bmatrix} \cos{300^{\circ}} & -\sin{300^{\circ}}\\ \sin{300^{\circ}} & \cos{300^{\circ}}\\ \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\\ \end{bmatrix}$. Then $C' = \begin{bmatrix} \frac{1}{2} & \frac{\sqrt{3}}{2}\\ -\frac{\sqrt{3}}{2} & \frac{1}{2}\\ \end{bmatrix} \cdot \begin{bmatrix} 1\\ 2\sqrt{3}\\ \end{bmatrix} = \begin{bmatrix} \frac{7}{2}\\ \frac{\sqrt{3}}{2}\\ \end{bmatrix}$. Thus, $C = (\frac{9}{2}, \frac{\sqrt{3}}{2})$. Since the triangle is equilateral, the center of the triangle is the average of the coordinates of the vertices. Then $P = (\frac{1 + 2 + \frac{9}{2}}{3}, \frac{2\sqrt{3} + \frac{\sqrt{3}}{2}}{3}) = (\frac{5}{2}, \frac{5\sqrt{3}}{6})$. Our answer is $\frac{5}{2} \cdot \frac{5\sqrt{3}}{6} = \frac{25\sqrt{3}}{12}$. $25 + 3 + 12 = \boxed{40}$

Solution 5

We construct point $C$ by drawing two circles with radius $r = AB = \sqrt{13}$. One circle will be centered at $A$, while the other is centered at $B$. The equations of the circles are:

$(x - 1)^2 + y^2 = 13$

$(x - 2)^2 + (y - 2\sqrt{3})^2 = 13$

Setting the LHS of each of these equations equal to each other and solving for $x$ yields after simplification:

$x = \frac{15}{2} - 2\sqrt{3}y$

Plugging that into the first equation gives the following quadratic in $y$ after simplification:

$y^2 - 2\sqrt{3}y + \frac{9}{4} = 0$

The quadratic formula gives $y = \frac{\sqrt{3}}{2}, \frac{3\sqrt{3}}{2}$.

Since $x > 0$ and $x = \frac{15}{2} - 2\sqrt{3}y$, we pick $y = \frac{\sqrt{3}}{2}$ in the hopes that it will give $x > 0$. Plugging $y$ into the equation for $x$ yields $x = \frac{9}{2}$.

Thus, $C(\frac{9}{2}, \frac{\sqrt{3}}{2})$. Averaging the coordinates of the vertices of equilateral triangle $ABC$ will give the center of mass of the triangle.

Thus, $P(\frac{5}{2}, \frac{5\sqrt{3}}{6})$, and the product of the coordinates is $\frac{25\sqrt{3}}{12}$, so the desired quantity is $\boxed{040}$.

Solution 6

Labeling our points and sketching a graph we get that $C$ is to the right of $AB$. Of course, we need to find $C$. Note that the transformation from $A$ to $B$ is $[1,2\sqrt{3}]$, and if we imagine a height dropped to $AB$ we see that a transformation from the midpoint $(\frac{3}{2},\sqrt {3})$ to $C$ is basically the first transformation, with $\frac{\sqrt{3}}{2}$ the magnitude and the x and y switched– then multiply the new y by -1. Then, applying this transformation of $[3,\frac{-\sqrt{3}}{2}]$ we get that $C=(\frac{9}{2},\frac{\sqrt{3}}{2})$ which means that $P=(\frac{5}{2},\frac{5\sqrt{3}}{6})$. Then our answer is $\boxed{40}$.

Solution 7

Transform this into the complex plane and let $a=1, b=2+2\sqrt3 i$. We know that 3 complex numbers $a,b,c$ form an equilateral triangle if $a^2+b^2+c^2=ab+bc+ac$, so plugging in our values of $a,b$, we get $8\sqrt3 i - 7 +c^2 = 2+2\sqrt3 i + (3+2\sqrt 3i)c.$ Solving for $c$ using Wolfram Alpha, we find that the solutions are $c=\frac 92 + \frac{i\sqrt3}{2}, -\frac 32 + \frac{3i\sqrt3}{2}$. The first one is in the first quadrant, so $C\left( \frac 92, \frac{\sqrt3}{2} \right)$. The center is the average of the coordinates and we find that it is $\left(\frac{5}2, \frac{5\sqrt3}{6} \right)$. Then $xy = \frac{25\sqrt3}{12} \implies 40$.

-bobthegod78, krwang, and Simplest14

See Also

2013 AIME II (Problems • Answer Key • Resources)
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