Difference between revisions of "2017 AMC 12A Problems/Problem 16"

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In the figure below, semicircles with centers at <math>A</math> and <math>B</math> and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter <math>JK</math>. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at <math>P</math> is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at <math>P</math>?
 
In the figure below, semicircles with centers at <math>A</math> and <math>B</math> and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter <math>JK</math>. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at <math>P</math> is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at <math>P</math>?
  
[[File:2017amc12a16.png|2017amc12a16.png]]
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[[File:2017amc12a16.png]]
  
 
<math> \textbf{(A)}\ 3/4
 
<math> \textbf{(A)}\ 3/4

Revision as of 18:47, 8 February 2017

Problem

In the figure below, semicircles with centers at $A$ and $B$ and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter $JK$. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at $P$ is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle. What is the radius of the circle centered at $P$?

2017amc12a16.png

$\textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 6/7 \qquad\textbf{(C)}\ 1/2 * sqrt3 \qquad\textbf{(D)}\ 5/8 * sqrt2 \qquad\textbf{(E)}\ 11/12$