MathDB

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

\overline{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

?
[asy] size(8cm); 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((-3,0)); dot((3,0)); dot(P); label("

J

",(-3,0),W); label("

A

",(-1,0),NW); label("

B

",(2,0),NE); label("

K

",(3,0),E); label("

P

",P,NW); [/asy]

<span class='latex-bold'>(A)</span>\ \frac{3}{4} \qquad <span class='latex-bold'>(B)</span>\ \frac{6}{7} \qquad<span class='latex-bold'>(C)</span>\ \frac{1}{2}\sqrt{3} \qquad<span class='latex-bold'>(D)</span>\ \frac{5}{8}\sqrt{2} \qquad<span class='latex-bold'>(E)</span>\ \frac{11}{12}

Problem Statement