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Difference between revisions of "1999 AMC 8 Problems/Problem 21"
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==Problem 21==
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The degree measure of angle <math>A</math> is
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<asy>
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unitsize(12);
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draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle);
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draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW));
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draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW));
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draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW));
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label(rotate(30)*"$40^\circ$",(2,-8.9),ENE);
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label("$100^\circ$",(21/3,-2/3),SE);
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label("$110^\circ$",(900/83,-317/83),NNW);
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label("$A$",(0,0),NW);
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</asy>
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<math>\text{(A)}\ 20 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 45</math>
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==Solution==
 
<asy>
 
<asy>
 
unitsize(12);
 
unitsize(12);

Revision as of 12:17, 17 June 2011

Problem 21

The degree measure of angle $A$ is

[asy] unitsize(12); draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle); draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW)); draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW)); draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW)); label(rotate(30)*"$40^\circ$",(2,-8.9),ENE); label("$100^\circ$",(21/3,-2/3),SE); label("$110^\circ$",(900/83,-317/83),NNW); label("$A$",(0,0),NW); [/asy]

$\text{(A)}\ 20 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 40 \qquad \text{(E)}\ 45$

Solution

[asy] unitsize(12); draw((0,0)--(20,0)--(1,-10)--(9,5)--(18,-8)--cycle); draw(arc((1,-10),(1+19/sqrt(461),-10+10/sqrt(461)),(25/17,-155/17),CCW)); draw(arc((19/3,0),(19/3-8/17,-15/17),(22/3,0),CCW)); draw(arc((900/83,-400/83),(900/83+19/sqrt(461),-400/83+10/sqrt(461)),(900/83 - 9/sqrt(97),-400/83 + 4/sqrt(97)),CCW)); label(rotate(30)*"$40^\circ$",(2,-8.9),ENE); label("$100^\circ$",(21/3,-2/3),SE); label("$110^\circ$",(900/83,-317/83),NNW); label("$A$",(0,0),NW); label("$B$", (20,0), NE); [/asy] Note that $\angle B=180-100-40=40^\circ$. So $\angle A=180-110-40=\boxed{30^\circ}$.

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