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>? | ||
| − | + | <asy> | |
| + | size(5cm); | ||
| + | draw(arc((0,0),3,0,180)); | ||
| + | draw(arc((2,0),1,0,180)); | ||
| + | draw(arc((-1,0),2,0,180)); | ||
| + | draw((-3,0)--(3,0)); | ||
| + | pair P = (-1,0)+(2+6/7)*dir(36.86989); | ||
| + | draw(circle(P,6/7)); | ||
| + | dot((-1,0)); dot((2,0)); dot(P); | ||
| + | </asy> | ||
| − | <math> \textbf{(A)}\ 3 | + | <math> \textbf{(A)}\ \frac{3}{4} |
| − | \qquad \textbf{(B)}\ 6 | + | \qquad \textbf{(B)}\ \frac{6}{7} |
| − | \qquad\textbf{(C)}\ 1 | + | \qquad\textbf{(C)}\ \frac{1}{2}\sqrt{3} |
| − | \qquad\textbf{(D)}\ 5 | + | \qquad\textbf{(D)}\ \frac{5}{8}\sqrt{2} |
| − | \qquad\textbf{(E)}\ 11 | + | \qquad\textbf{(E)}\ \frac{11}{12} </math> |
Revision as of 18:48, 8 February 2017
Problem
In the figure below, semicircles with centers at 
and 
and with radii 2 and 1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter 
. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at 
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 ?
![[asy] size(5cm); draw(arc((0,0),3,0,180)); draw(arc((2,0),1,0,180)); draw(arc((-1,0),2,0,180)); draw((-3,0)--(3,0)); pair P = (-1,0)+(2+6/7)*dir(36.86989); draw(circle(P,6/7)); dot((-1,0)); dot((2,0)); dot(P); [/asy]](http://latex.artofproblemsolving.com/c/7/0/c70afdd05a6954478364e08a7b13089dc5022e4a.png)
