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Difference between revisions of "1972 AHSME Problems/Problem 30"

(Created page with "== Problem 30 == == Solution == Let the rectangle <math>ABCD</math> have crease <math>BE</math> with <math>E</math> on <math>CD</math>, and let <math>F</math> be on <math>A...")
 
 
(4 intermediate revisions by 2 users not shown)
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== Problem 30 ==
 
== Problem 30 ==
  
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<asy>
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real h = 7;
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real t = asin(6/h)/2;
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real x = 6-h*tan(t);
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real y = x*tan(2*t);
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draw((0,0)--(0,h)--(6,h)--(x,0)--cycle);
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draw((x,0)--(0,y)--(6,h));
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draw((6,h)--(6,0)--(x,0),dotted);
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label("L",(3.75,h/2),W);
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label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm));
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label("6''",(3,0),S);
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draw((2.5,-.5)--(0,-.5),Arrow(2mm));
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draw((3.5,-.5)--(6,-.5),Arrow(2mm));
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draw((0,-.25)--(0,-.75));draw((6,-.25)--(6,-.75));
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//Credit to Zimbalono for the diagram
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</asy>
  
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A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle <math>\theta</math> is
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<math>\textbf{(A) }3\sec ^2\theta\csc\theta\qquad
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\textbf{(B) }6\sin\theta\sec\theta\qquad
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\textbf{(C) }3\sec\theta\csc\theta\qquad
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\textbf{(D) }6\sec\theta\csc^2\theta\qquad
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\textbf{(E) }\text{None of these}    </math>
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== Solution ==
 
== Solution ==
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<asy>
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real h = 7;
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real t = asin(6/h)/2;
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real x = 6-h*tan(t);
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real y = x*tan(2*t);
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draw((0,0)--(0,h)--(6,h)--(x,0)--cycle);
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draw((x,0)--(0,y)--(6,h));
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draw((6,h)--(6,0)--(x,0),dotted);
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label("A",(0,h),NW);label("B",(6,h),NE);label("C",(6,0),SE);label("D",(0,0),SW);label("E",(x,0),N);label("F",(0,y),W);
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label("L",(3.75,h/2),W);
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label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm));
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label("6''",(3,0),S);
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draw((2.5,-.5)--(0,-.5),Arrow(2mm));
 +
draw((3.5,-.5)--(6,-.5),Arrow(2mm));
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draw((0,-.25)--(0,-.75));draw((6,-.25)--(6,-.75));
 +
//Credit to Zimbalono for the diagram
 +
</asy>
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Let the rectangle <math>ABCD</math> have crease <math>BE</math> with <math>E</math> on <math>CD</math>, and let <math>F</math> be on <math>AD</math> such that <math>F</math> is a reflection of <math>C</math> over <math>BE</math>.  Notice that triangles <math>ABF</math> and <math>DEF</math> are similar, so by setting <math>CE = EF = x</math> with <math>DE = 6-x</math>, giving <math>DF = 2\sqrt{3x-9}</math> we have that <math>AF = \frac{18-3x}{\sqrt{3x-9}}</math>. Noticing that <math>BC = BF = x\cot{\theta}</math> gives <math>\frac{(18-3x)^2}{3x-9}+36 = x^2\cot^2{\theta} \Rightarrow \frac{3(x-6)^2}{x-3}+36 = \frac{3x^2}{x-3} = x^2\cot^2{\theta}</math>. <math>x = \frac{3\cot^2{\theta}+3}{\cot^2{\theta}} = \frac{3\csc^2{\theta}}{\cot^2{\theta}} = 3\sec^2{\theta}</math>. Noticing that <math>BE = x\sqrt{\cot^2{\theta}+1} = x\csc{\theta}</math> gives the answer to be <math>3\sec^2{\theta}\csc{\theta}</math>.
  
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Let the rectangle <math>ABCD</math> have crease <math>BE</math> with <math>E</math> on <math>CD</math>, and let <math>F</math> be on <math>AD</math> such that <math>F</math> is a reflection of <math>C</math> over <math>BE</math>.  Notice that triangles <math>ABF</math> and <math>DEF</math> are similar, so by setting <math>CE = EF = x</math> with <math>DE = 6-x</math>, giving <math>DF = 2\sqrt{3x-9}</math> we have that <math>AF = \frac{18-3x}{\sqrt{3x-9}}</math>. Noticing that <math>BC = BF = x\tan{\theta}</math> gives <math>\frac{(18-3x)^2}{3x-9}+36 = x^2\tan^2{\theta} \Rightarrow \frac{3(x-6)^2}{x-3}+36 = \frac{3x^2}{x-3} = x^2\tan^2{\theta}</math>. <math>x = \frac{3\tan^2{\theta}+3}{\tan^2{\theta}} = \frac{3\sec^2{\theta}}{\tan^2{\theta}} = 3\csc^2{\theta}</math>. Noticing that <math>BE = x\sqrt{\tan^2{\theta}+1} = x\sec{\theta}</math> gives the answer to be <math>3\csc^2{\theta}\sec{\theta}</math> which is not in the answer list?
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Thanks to xiej for correcting my mistake!
  
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If anyone could correct my solution, please help and thanks!
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==See Also==
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P.S correct answer is <math>3\sec^2{\theta}\csc{\theta}</math>.
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{{AHSME box|year=1972|num-b=29|num-a=31}}
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{{MAA Notice}}

Latest revision as of 18:52, 23 June 2021

Problem 30

[asy] real h = 7; real t = asin(6/h)/2; real x = 6-h*tan(t); real y = x*tan(2*t); draw((0,0)--(0,h)--(6,h)--(x,0)--cycle); draw((x,0)--(0,y)--(6,h)); draw((6,h)--(6,0)--(x,0),dotted); label("L",(3.75,h/2),W); label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm)); label("6''",(3,0),S); draw((2.5,-.5)--(0,-.5),Arrow(2mm)); draw((3.5,-.5)--(6,-.5),Arrow(2mm)); draw((0,-.25)--(0,-.75));draw((6,-.25)--(6,-.75)); //Credit to Zimbalono for the diagram [/asy]

A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle $\theta$ is

$\textbf{(A) }3\sec ^2\theta\csc\theta\qquad \textbf{(B) }6\sin\theta\sec\theta\qquad \textbf{(C) }3\sec\theta\csc\theta\qquad \textbf{(D) }6\sec\theta\csc^2\theta\qquad \textbf{(E) }\text{None of these}$

Solution

[asy] real h = 7; real t = asin(6/h)/2; real x = 6-h*tan(t); real y = x*tan(2*t); draw((0,0)--(0,h)--(6,h)--(x,0)--cycle); draw((x,0)--(0,y)--(6,h)); draw((6,h)--(6,0)--(x,0),dotted); label("A",(0,h),NW);label("B",(6,h),NE);label("C",(6,0),SE);label("D",(0,0),SW);label("E",(x,0),N);label("F",(0,y),W); label("L",(3.75,h/2),W); label("$\theta$",(6,h-1.5),W);draw(arc((6,h),2,270,270-degrees(t)),Arrow(2mm)); label("6''",(3,0),S); draw((2.5,-.5)--(0,-.5),Arrow(2mm)); draw((3.5,-.5)--(6,-.5),Arrow(2mm)); draw((0,-.25)--(0,-.75));draw((6,-.25)--(6,-.75)); //Credit to Zimbalono for the diagram [/asy] Let the rectangle $ABCD$ have crease $BE$ with $E$ on $CD$, and let $F$ be on $AD$ such that $F$ is a reflection of $C$ over $BE$. Notice that triangles $ABF$ and $DEF$ are similar, so by setting $CE = EF = x$ with $DE = 6-x$, giving $DF = 2\sqrt{3x-9}$ we have that $AF = \frac{18-3x}{\sqrt{3x-9}}$. Noticing that $BC = BF = x\cot{\theta}$ gives $\frac{(18-3x)^2}{3x-9}+36 = x^2\cot^2{\theta} \Rightarrow \frac{3(x-6)^2}{x-3}+36 = \frac{3x^2}{x-3} = x^2\cot^2{\theta}$. $x = \frac{3\cot^2{\theta}+3}{\cot^2{\theta}} = \frac{3\csc^2{\theta}}{\cot^2{\theta}} = 3\sec^2{\theta}$. Noticing that $BE = x\sqrt{\cot^2{\theta}+1} = x\csc{\theta}$ gives the answer to be $3\sec^2{\theta}\csc{\theta}$.

Thanks to xiej for correcting my mistake!

See Also

1972 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 29 		Followed by
Problem 31
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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