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| − | ==Problem==
| + | #redirect [[2004 AMC 12B Problems/Problem 10]] |
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| − | An annulus is the region between two concentric circles. The concentric circles in the figure have radii <math>b</math> and <math>c</math>, with <math>b>c</math>. Let <math>OX</math> be a radius of the larger circle, let <math>XZ</math> be tangent to the smaller circle at <math>Z</math>, and let <math>OY</math> be the radius of the larger circle that contains <math>Z</math>. Let <math>a=XZ</math>, <math>d=YZ</math>, and <math>e=XY</math>. What is the area of the annulus?
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| − | <asy>
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| − | unitsize(1.5cm);
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| − | defaultpen(0.8);
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| − | real r1=1.5, r2=2.5;
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| − | pair O=(0,0);
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| − | path inner=Circle(O,r1), outer=Circle(O,r2);
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| − | pair Y=(0,r2), Z=(0,r1), X=intersectionpoint( Z--(Z+(10,0)), outer );
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| − | filldraw(outer,lightgray,black);
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| − | filldraw(inner,white,black);
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| − | draw(X--O--Y); draw(Y--X--Z);
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| − | label("$O$",O,SW);
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| − | label("$X$",X,E);
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| − | label("$Y$",Y,N);
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| − | label("$Z$",Z,SW);
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| − | label("$a$",X--Z,N);
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| − | label("$b$",0.25*X,SE);
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| − | label("$c$",O--Z,E);
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| − | label("$d$",Y--Z,W);
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| − | label("$e$",Y*0.65 + X*0.35,SW);
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| − | defaultpen(0.5);
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| − | dot(O); dot(X); dot(Z); dot(Y);
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| − | </asy>
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| − | <math> \mathrm{(A) \ } \pi a^2 \qquad \mathrm{(B) \ } \pi b^2 \qquad \mathrm{(C) \ } \pi c^2 \qquad \mathrm{(D) \ } \pi d^2 \qquad \mathrm{(E) \ } \pi e^2 </math>
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| − | ==Solution==
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| − | The area of the large circle is <math>\pi b^2</math>, the area of the small one is <math>\pi a^2</math>, hence the shaded area is <math>\pi(b^2-c^2)</math>.
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| − | From the [[Pythagorean Theorem]] for the right triangle <math>OXZ</math> we have <math>a^2 + c^2 = b^2</math>, hence <math>b^2-c^2=a^2</math> and thus the shaded area is <math>\boxed{\pi a^2}</math>.
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| − | == See also ==
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| − | {{AMC10 box|year=2004|ab=B|num-b=11|num-a=13}}
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