https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Epic-One&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T12:17:37ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2012_AMC_10A_Problems/Problem_11&diff=711532012 AMC 10A Problems/Problem 112015-07-14T14:52:16Z<p>Epic-One: /* Solution */</p>
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<div>== Problem ==<br />
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Externally tangent circles with centers at points A and B have radii of lengths 5 and 3, respectively. A line externally tangent to both circles intersects ray AB at point C. What is BC?<br />
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<math> \textbf{(A)}\ 4\qquad\textbf{(B)}\ 4.8\qquad\textbf{(C)}\ 10.2\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 14.4 </math><br />
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== Solution ==<br />
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<center><asy><br />
unitsize(3.5mm);<br />
defaultpen(linewidth(.8pt)+fontsize(10pt));<br />
dotfactor=4;<br />
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pair A=(0,0), B=(8,0); pair C=(20,0); pair D=(1.25,-0.25sqrt(375)); pair E=(8.75,-0.15sqrt(375));<br />
path a=Circle(A,5);<br />
path b=Circle(B,3);<br />
draw(a); draw(b);<br />
draw(C--D);<br />
draw(A--C);<br />
draw(A--D);<br />
draw(B--E);<br />
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pair[] ps={A,B,C,D,E};<br />
dot(ps);<br />
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label("$A$",A,N); label("$B$",B,N); label("$C$",C,N);<br />
label("$D$",D,SE); label("$E$",E,SE);<br />
label("$5$",(A--D),SW);<br />
label("$3$",(B--E),SW);<br />
label("$8$",(A--B),N);<br />
label("$x$",(C--B),N);<br />
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</asy></center><br />
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Let <math>D</math> and <math>E</math> be the points of tangency on circles <math>A</math> and <math>B</math> with line <math>CD</math>. <math>AB=8</math>. Also, let <math>BC=x</math>. As <math>\angle ADC</math> and <math>\angle BEC</math> are right angles (a radius is perpendicular to a tangent line at the point of tangency) and both triangles share <math>\angle ACD</math>, <math>\triangle ADC \sim \triangle BEC</math>. From this we can get a proportion.<br />
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<math>\frac{BC}{AC}=\frac{BE}{AD} \rightarrow \frac{x}{x+8}=\frac{3}{5} \rightarrow 5x=3x+24 \rightarrow x=\boxed{\textbf{(D)}\ 12}</math><br />
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== See Also ==<br />
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{{AMC10 box|year=2012|ab=A|num-b=10|num-a=12}}<br />
{{MAA Notice}}</div>Epic-Onehttps://artofproblemsolving.com/wiki/index.php?title=User:Epic-One&diff=59588User:Epic-One2014-02-12T04:11:44Z<p>Epic-One: Created page with "Hello World"</p>
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<div>Hello World</div>Epic-One