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Difference between revisions of "2014 AMC 12B Problems/Problem 21"

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==Problem 21==
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== Problem ==
 +
In the figure, <math> ABCD </math> is a square of side length <math> 1 </math>. The rectangles <math> JKHG </math> and <math> EBCF </math> are congruent. What is <math> BE </math>?
  
−
In the figure, <math> ABCD </math> is a square of side length <math> 1 </math>. The rectangles <math> JKHG </math> and <math> EBCF </math> are congruent. What is <math> BE </math>?
 
 
<asy>
 
<asy>
 
pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2);
 
pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2);
Line 10: Line 10:
 
label("$G$",G,E); label("$H$",H,N); label("$J$",J,S); label("$K$",K,W);
 
label("$G$",G,E); label("$H$",H,N); label("$J$",J,S); label("$K$",K,W);
 
</asy>
 
</asy>
 +
 
<math> \textbf{(A) }\frac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\frac{\sqrt{3}}{6}\qquad\textbf{(E) } 1-\frac{\sqrt{2}}{2}</math>
 
<math> \textbf{(A) }\frac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\frac{\sqrt{3}}{6}\qquad\textbf{(E) } 1-\frac{\sqrt{2}}{2}</math>
  
−
==Solution==
+
== Solution 1 ==
 +
Draw the altitude from <math>H</math> to <math>AB</math> and call the foot <math>L</math>. Then <math>HL=1</math>. Consider <math>HJ</math>. It is the hypotenuse of both right triangles <math>\triangle HGJ</math> and <math>\triangle HLJ</math>, and we know <math>JG=HL=1</math>, so we must have <math>\triangle HGJ\cong\triangle JLH</math> by Hypotenuse-Leg congruence. From this congruence we have <math>LJ=HG=BE</math>.
 +
 
 +
Notice that all four triangles in this picture are similar. Also, we have <math>LA=HD=EJ</math>. So set <math>x=LJ=HG=BE</math> and <math>y=LA=HD=EJ</math>. Now <math>BE+EJ+JL+LA=2(x+y)=1</math>. This means <math>x+y=\frac{1}{2}=BE+EJ=BJ</math>, so <math>J</math> is the midpoint of <math>AB</math>. So <math>\triangle AJG</math>, along with all other similar triangles in the picture, is a 30-60-90 triangle, and we have <math>AG=\sqrt{3} \cdot AJ=\sqrt{3}/2</math> and subsequently <math>GD=\frac{2-\sqrt{3}}{2}=KE</math>. This means <math>EJ=\sqrt{3} \cdot KE=\frac{2\sqrt{3}-3}{2}</math>, which gives <math>BE=\frac{1}{2}-EJ=\frac{4-2\sqrt{3}}{2}=2-\sqrt{3}</math>, so the answer is <math>\textbf{(C)}</math>.
 +
 
 +
== Solution 2 ==
 
Let <math>BE = x</math>.  Let <math>JA = y</math>.  Because <math>\angle FKH = \angle EJK = \angle AGJ = \angle DHG</math> and <math>\angle FHK = \angle EKJ = \angle AJG = \angle DGH</math>, <math>\triangle KEJ, \triangle JAG, \triangle GDH, \triangle HFK</math> are all similar.  Using proportions and the pythagorean theorem, we find  
 
Let <math>BE = x</math>.  Let <math>JA = y</math>.  Because <math>\angle FKH = \angle EJK = \angle AGJ = \angle DHG</math> and <math>\angle FHK = \angle EKJ = \angle AJG = \angle DGH</math>, <math>\triangle KEJ, \triangle JAG, \triangle GDH, \triangle HFK</math> are all similar.  Using proportions and the pythagorean theorem, we find  
 
<cmath>EK = xy</cmath>
 
<cmath>EK = xy</cmath>
 
<cmath>FK = \sqrt{1-y^2}</cmath>
 
<cmath>FK = \sqrt{1-y^2}</cmath>
 
<cmath>EJ = x\sqrt{1-y^2}</cmath>
 
<cmath>EJ = x\sqrt{1-y^2}</cmath>
−
Because we know that <math>BE+EJ+AJ = EK + FK = 1</math>, we can set up a systems of equations
+
Because we know that <math>BE+EJ+AJ = EK + FK = 1</math>, we can set up a systems of equations and solving for <math>x</math>, we get
−
<cmath>x + x\sqrt{1-y^2} + y = 1</cmath>
+
<cmath>x + x\sqrt{1-y^2} + y = 1 \implies x= \frac{1-y}{1+\sqrt{1-y^2}}</cmath>
−
<cmath>xy + \sqrt{1-y^2} = 1</cmath>
+
<cmath>xy + \sqrt{1-y^2} = 1 \implies x= \frac{1-\sqrt{1-y^2}}{y}</cmath>
−
Solving for <math>x</math> in the second equation, we get
+
Now solving for <math>y</math>, we get  
−
<cmath>x= \frac{1-\sqrt{1-y^2}}{y}</cmath>
+
<cmath>\frac{1-y}{1+\sqrt{1-y^2}}=\frac{1-\sqrt{1-y^2}}{y} \implies y(1-y)=(1-\sqrt{1-y^2})(1+\sqrt{1-y^2}) \implies y-y^2=y^2 \implies y=\frac{1}{2}</cmath>
−
Plugging this into the first equation, we get
+
Plugging into the second equations with <math>x</math>, we get
−
<cmath>\frac{1-\sqrt{1-y^2}}{y} + (\sqrt{1-y^2})\frac{1-\sqrt{1-y^2}}{y} + y = 1 \implies \frac{2y^2}{y}=1 \implies y=\frac{1}{2}</cmath>
 
−
Plugging into the previous equation with <math>x</math>, we get
 
 
<cmath>x= 2\left(1-\sqrt{1-\frac{1}{4}}\right) = 2\left(\frac{2-\sqrt{3}}{2} \right) = \boxed{\textbf{(C)}\ 2-\sqrt{3}}</cmath>
 
<cmath>x= 2\left(1-\sqrt{1-\frac{1}{4}}\right) = 2\left(\frac{2-\sqrt{3}}{2} \right) = \boxed{\textbf{(C)}\ 2-\sqrt{3}}</cmath>
 +
 +
== Solution 3 ==
 +
Let <math>BE = x</math>, <math>EK = a</math>, and <math>EJ = b</math>. Then <math>x^2 = a^2 + b^2</math> and because <math>\triangle KEJ \cong \triangle GDH</math> and <math>\triangle KEJ \sim \triangle JAG</math>, <math>\frac{GA}{1} = 1 - a = \frac{b}{x}</math>. Furthermore, the area of the four triangles and the two rectangles sums to 1:
 +
 +
<cmath>1 = 2x + GA\cdot JA + ab</cmath>
 +
 +
<cmath>1 = 2x + (1 - a)(1 - (x + b)) + ab</cmath>
 +
 +
<cmath>1 = 2x + \frac{b}{x}(1 - x - b) + \left(1 - \frac{b}{x}\right)b</cmath>
 +
 +
<cmath>1 = 2x + \frac{b}{x} - b - \frac{b^2}{x} + b - \frac{b^2}{x}</cmath>
 +
 +
<cmath>x = 2x^2 + b - 2b^2</cmath>
 +
 +
<cmath>x - b = 2(x - b)(x + b)</cmath>
 +
 +
<cmath>x + b = \frac{1}{2}</cmath>
 +
 +
<cmath>b = \frac{1}{2} - x</cmath>
 +
 +
<cmath>a = 1 - \frac{b}{x} = 2 - \frac{1}{2x}</cmath>
 +
 +
By the Pythagorean theorem: <math>x^2 = a^2 + b^2</math>
 +
 +
<cmath>x^2 = \left(2 - \frac{1}{2x}\right)^2 + \left(\frac{1}{2} - x\right)^2</cmath>
 +
 +
<cmath>x^2 = 4 - \frac{2}{x} + \frac{1}{4x^2} + \frac{1}{4} - x + x^2</cmath>
 +
 +
<cmath>0 =  \frac{1}{4x^2} - \frac{2}{x} + \frac{17}{4} - x</cmath>
 +
 +
<cmath>0 = 1 - 8x + 17x^2 - 4x^3.</cmath>
 +
 +
Then by the rational root theorem, this has roots <math>\frac{1}{4}</math>, <math>2 - \sqrt{3}</math>, and <math>2 + \sqrt{3}</math>. The first and last roots are extraneous because they imply <math>a = 0</math> and <math>x > 1</math>, respectively, thus <math>x = \boxed{\textbf{(C)}\ 2-\sqrt{3}}</math>.
 +
 +
== Solution 4 ==
 +
Let <math>\angle FKH</math> = <math>k</math> and <math>CF</math> = <math>a</math>.  It is shown that all four triangles in the picture are similar.  From the square side lengths:
 +
 +
<cmath>a + \sin(k) \cdot 1 + \cos(k) \cdot a = 1</cmath>
 +
 +
<cmath>\sin(k)a + \cos(k) = 1</cmath>
 +
Solving for <math>a</math> we get:
 +
 +
<cmath>a = \frac{1-\sin(k)}{\cos(k) + 1} = \frac{1 - \cos(k)}{\sin(k)}</cmath>
 +
 +
<cmath>(1-\sin(k)) \cdot sin(k) = (1 - \cos(k))\cdot(\cos(k) + 1)</cmath>
 +
 +
<cmath>\sin(k)-\sin(k)^2 = \cos(k) + 1 - \cos(k)^2 - \cos(k)</cmath>
 +
 +
<cmath>\sin(k)-\sin(k)^2 = \sin(k)^2</cmath>
 +
 +
<cmath>1-\sin(k) = \sin(k)</cmath>
 +
 +
<cmath>\sin(k) = \frac{1}{2}, \cos(k) = \frac{\sqrt 3}{2}</cmath>
 +
 +
<cmath>a = \frac{1 - \frac{\sqrt 3}{2}}{\frac{1}{2}} = 2 - \sqrt 3</cmath>
 +
 +
== Solution 5 ==
 +
Note that <math>HJ</math> is a diagonal of <math>JKHG</math>, so it must be equal in length to <math>FB</math>. Therefore, quadrilateral <math>FHJB</math> has <math>FH\parallel BJ</math>, and <math>FB=HJ</math>, so it must be either an isosceles trapezoid or a parallelogram. But due to the slope of <math>FB</math> and <math>HJ</math>, we see that it must be a parallelogram. Therefore, <math>FH=BJ</math>. But by the symmetry in rectangle <math>FEAD</math>, we see that <math>FH=JA</math>. Therefore, <math>BJ=FH=JA</math>. We also know that <math>BJ+JA=1</math>, hence <math>BJ=JA=\frac12</math>.
 +
 +
As <math>JG=1</math> and <math>JA=\frac12</math>, and as <math>\triangle GJA</math> is right, we know that <math>\triangle GJA</math> must be a 30-60-90 triangle. Therefore, <math>GA=\sqrt{3}/2</math> and <math>DG=1-\sqrt{3}/2</math>. But by similarity, <math>\triangle DHG</math> is also a 30-60-90 triangle, hence <math>DH=\sqrt{3}-3/2</math>. But <math>\triangle DHG\cong\triangle EJK</math>, hence <math>EJ=\sqrt{3}-3/2</math>. As <math>BJ=1/2</math>, this implies that <math>BE=BJ-EJ=1/2-\sqrt{3}+3/2=2-\sqrt{3}</math>. Thus the answer is <math>\boxed{\textbf{(C)}}</math>.
 +
 +
== Solution 6 ==
 +
 +
Let <math>BE = x</math>, <math>KE = y</math>
 +
 +
<math>DG = y</math>, <math>AG = 1-y</math>, <math>KJ = x</math>, <math>JG = 1</math>
 +
 +
<math>\frac{AJ}{JG} = \frac{KE}{JK}</math>, <math>AJ = \frac{y}{x}</math>, <math>EJ = 1 - x - \frac{y}{x}</math>,
 +
 +
<math>\frac{AG}{JG} = \frac{EJ}{JK}</math>, <math>1-y = \frac{1 - x - \frac{y}{x}}{x}</math>, <math>x - xy = 1 - x - \frac{y}{x}</math>, <math>2x^2 - x^2y + y - x = 0</math> <math>(1)</math>
 +
 +
<math>AJ^2 + AG^2 = JG^2</math>, <math>\left( \frac{y}{x} \right) ^2 + (1-y)^2 = 1</math>, <math>\frac{y^2}{x^2} + 1 - 2y + y^2 = 1</math>, <math>x^2 y -2x^2 + y = 0</math> <math>(2)</math>
 +
 +
<math>(1) + (2)</math>, <math>2y - x = 0</math>, <math>y = \frac{x}{2}</math>
 +
 +
Substitute <math>y</math> into <math>(2)</math> we get <math>x^2 \cdot \frac{x}{2} - 2 x^2 + \frac{x}{2} = 0</math>, <math>x^2 - 4x + 1 = 0</math>
 +
 +
<math>x = \frac{4 \pm \sqrt{16-4}}{2} = 2 \pm \sqrt{3}</math>, as <math>x < 1</math>, <math>x = \boxed{\textbf{(C)}2 - \sqrt{3}}</math>
 +
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 +
 +
== See Also ==
 +
{{AMC12 box|year=2014|ab=B|num-b=20|num-a=22}}
 +
{{MAA Notice}}
 +
[[Category: Introductory Geometry Problems]]

Latest revision as of 01:07, 1 January 2023

Problem

In the figure, $ABCD$ is a square of side length $1$. The rectangles $JKHG$ and $EBCF$ are congruent. What is $BE$?

[asy] pair A=(1,0), B=(0,0), C=(0,1), D=(1,1), E=(2-sqrt(3),0), F=(2-sqrt(3),1), G=(1,sqrt(3)/2), H=(2.5-sqrt(3),1), J=(.5,0), K=(2-sqrt(3),1-sqrt(3)/2); draw(A--B--C--D--cycle); draw(K--H--G--J--cycle); draw(F--E); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,NW); label("$D$",D,NE); label("$E$",E,S); label("$F$",F,N); label("$G$",G,E); label("$H$",H,N); label("$J$",J,S); label("$K$",K,W); [/asy]

$\textbf{(A) }\frac{1}{2}(\sqrt{6}-2)\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }2-\sqrt{3}\qquad\textbf{(D) }\frac{\sqrt{3}}{6}\qquad\textbf{(E) } 1-\frac{\sqrt{2}}{2}$

Solution 1

Draw the altitude from $H$ to $AB$ and call the foot $L$. Then $HL=1$. Consider $HJ$. It is the hypotenuse of both right triangles $\triangle HGJ$ and $\triangle HLJ$, and we know $JG=HL=1$, so we must have $\triangle HGJ\cong\triangle JLH$ by Hypotenuse-Leg congruence. From this congruence we have $LJ=HG=BE$.

Notice that all four triangles in this picture are similar. Also, we have $LA=HD=EJ$. So set $x=LJ=HG=BE$ and $y=LA=HD=EJ$. Now $BE+EJ+JL+LA=2(x+y)=1$. This means $x+y=\frac{1}{2}=BE+EJ=BJ$, so $J$ is the midpoint of $AB$. So $\triangle AJG$, along with all other similar triangles in the picture, is a 30-60-90 triangle, and we have $AG=\sqrt{3} \cdot AJ=\sqrt{3}/2$ and subsequently $GD=\frac{2-\sqrt{3}}{2}=KE$. This means $EJ=\sqrt{3} \cdot KE=\frac{2\sqrt{3}-3}{2}$, which gives $BE=\frac{1}{2}-EJ=\frac{4-2\sqrt{3}}{2}=2-\sqrt{3}$, so the answer is $\textbf{(C)}$.

Solution 2

Let $BE = x$. Let $JA = y$. Because $\angle FKH = \angle EJK = \angle AGJ = \angle DHG$ and $\angle FHK = \angle EKJ = \angle AJG = \angle DGH$, $\triangle KEJ, \triangle JAG, \triangle GDH, \triangle HFK$ are all similar. Using proportions and the pythagorean theorem, we find \[EK = xy\] \[FK = \sqrt{1-y^2}\] \[EJ = x\sqrt{1-y^2}\] Because we know that $BE+EJ+AJ = EK + FK = 1$, we can set up a systems of equations and solving for $x$, we get \[x + x\sqrt{1-y^2} + y = 1 \implies x= \frac{1-y}{1+\sqrt{1-y^2}}\] \[xy + \sqrt{1-y^2} = 1 \implies x= \frac{1-\sqrt{1-y^2}}{y}\] Now solving for $y$, we get \[\frac{1-y}{1+\sqrt{1-y^2}}=\frac{1-\sqrt{1-y^2}}{y} \implies y(1-y)=(1-\sqrt{1-y^2})(1+\sqrt{1-y^2}) \implies y-y^2=y^2 \implies y=\frac{1}{2}\] Plugging into the second equations with $x$, we get \[x= 2\left(1-\sqrt{1-\frac{1}{4}}\right) = 2\left(\frac{2-\sqrt{3}}{2} \right) = \boxed{\textbf{(C)}\ 2-\sqrt{3}}\]

Solution 3

Let $BE = x$, $EK = a$, and $EJ = b$. Then $x^2 = a^2 + b^2$ and because $\triangle KEJ \cong \triangle GDH$ and $\triangle KEJ \sim \triangle JAG$, $\frac{GA}{1} = 1 - a = \frac{b}{x}$. Furthermore, the area of the four triangles and the two rectangles sums to 1:

\[1 = 2x + GA\cdot JA + ab\]

\[1 = 2x + (1 - a)(1 - (x + b)) + ab\]

\[1 = 2x + \frac{b}{x}(1 - x - b) + \left(1 - \frac{b}{x}\right)b\]

\[1 = 2x + \frac{b}{x} - b - \frac{b^2}{x} + b - \frac{b^2}{x}\]

\[x = 2x^2 + b - 2b^2\]

\[x - b = 2(x - b)(x + b)\]

\[x + b = \frac{1}{2}\]

\[b = \frac{1}{2} - x\]

\[a = 1 - \frac{b}{x} = 2 - \frac{1}{2x}\]

By the Pythagorean theorem: $x^2 = a^2 + b^2$

\[x^2 = \left(2 - \frac{1}{2x}\right)^2 + \left(\frac{1}{2} - x\right)^2\]

\[x^2 = 4 - \frac{2}{x} + \frac{1}{4x^2} + \frac{1}{4} - x + x^2\]

\[0 = \frac{1}{4x^2} - \frac{2}{x} + \frac{17}{4} - x\]

\[0 = 1 - 8x + 17x^2 - 4x^3.\]

Then by the rational root theorem, this has roots $\frac{1}{4}$, $2 - \sqrt{3}$, and $2 + \sqrt{3}$. The first and last roots are extraneous because they imply $a = 0$ and $x > 1$, respectively, thus $x = \boxed{\textbf{(C)}\ 2-\sqrt{3}}$.

Solution 4

Let $\angle FKH$ = $k$ and $CF$ = $a$. It is shown that all four triangles in the picture are similar. From the square side lengths:

\[a + \sin(k) \cdot 1 + \cos(k) \cdot a = 1\]

\[\sin(k)a + \cos(k) = 1\] Solving for $a$ we get:

\[a = \frac{1-\sin(k)}{\cos(k) + 1} = \frac{1 - \cos(k)}{\sin(k)}\]

\[(1-\sin(k)) \cdot sin(k) = (1 - \cos(k))\cdot(\cos(k) + 1)\]

\[\sin(k)-\sin(k)^2 = \cos(k) + 1 - \cos(k)^2 - \cos(k)\]

\[\sin(k)-\sin(k)^2 = \sin(k)^2\]

\[1-\sin(k) = \sin(k)\]

\[\sin(k) = \frac{1}{2}, \cos(k) = \frac{\sqrt 3}{2}\]

\[a = \frac{1 - \frac{\sqrt 3}{2}}{\frac{1}{2}} = 2 - \sqrt 3\]

Solution 5

Note that $HJ$ is a diagonal of $JKHG$, so it must be equal in length to $FB$. Therefore, quadrilateral $FHJB$ has $FH\parallel BJ$, and $FB=HJ$, so it must be either an isosceles trapezoid or a parallelogram. But due to the slope of $FB$ and $HJ$, we see that it must be a parallelogram. Therefore, $FH=BJ$. But by the symmetry in rectangle $FEAD$, we see that $FH=JA$. Therefore, $BJ=FH=JA$. We also know that $BJ+JA=1$, hence $BJ=JA=\frac12$.

As $JG=1$ and $JA=\frac12$, and as $\triangle GJA$ is right, we know that $\triangle GJA$ must be a 30-60-90 triangle. Therefore, $GA=\sqrt{3}/2$ and $DG=1-\sqrt{3}/2$. But by similarity, $\triangle DHG$ is also a 30-60-90 triangle, hence $DH=\sqrt{3}-3/2$. But $\triangle DHG\cong\triangle EJK$, hence $EJ=\sqrt{3}-3/2$. As $BJ=1/2$, this implies that $BE=BJ-EJ=1/2-\sqrt{3}+3/2=2-\sqrt{3}$. Thus the answer is $\boxed{\textbf{(C)}}$.

Solution 6

Let $BE = x$, $KE = y$

$DG = y$, $AG = 1-y$, $KJ = x$, $JG = 1$

$\frac{AJ}{JG} = \frac{KE}{JK}$, $AJ = \frac{y}{x}$, $EJ = 1 - x - \frac{y}{x}$,

$\frac{AG}{JG} = \frac{EJ}{JK}$, $1-y = \frac{1 - x - \frac{y}{x}}{x}$, $x - xy = 1 - x - \frac{y}{x}$, $2x^2 - x^2y + y - x = 0$ $(1)$

$AJ^2 + AG^2 = JG^2$, $\left( \frac{y}{x} \right) ^2 + (1-y)^2 = 1$, $\frac{y^2}{x^2} + 1 - 2y + y^2 = 1$, $x^2 y -2x^2 + y = 0$ $(2)$

$(1) + (2)$, $2y - x = 0$, $y = \frac{x}{2}$

Substitute $y$ into $(2)$ we get $x^2 \cdot \frac{x}{2} - 2 x^2 + \frac{x}{2} = 0$, $x^2 - 4x + 1 = 0$

$x = \frac{4 \pm \sqrt{16-4}}{2} = 2 \pm \sqrt{3}$, as $x < 1$, $x = \boxed{\textbf{(C)}2 - \sqrt{3}}$

~isabelchen

See Also

2014 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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All AMC 12 Problems and Solutions

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