Question

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 $P?$

$\dfrac{3}{4}$

$\dfrac{6}{7}$

$\dfrac{1}{2}\sqrt3$

$\dfrac{5}{8}\sqrt2$

$\dfrac{11}{12}$

Solution:

The large semicircle has radius $3$ and center $C,$ the midpoint of $\overline{JK}.$ Placing $J$ at the origin, $A=2,$ $B=5,$ $C=3,$ $K=6$ along the base. Let $r$ be the radius of the circle at $P.$

By tangency, $PA=2+r,$ $PB=1+r,$ and $PC=3-r.$ Dropping a perpendicular from $P$ to the base at horizontal position $3+x$ with height $h,$ the Pythagorean theorem gives $h^2=(2+r)^2-(1+x)^2=(3-r)^2-x^2=(1+r)^2-(2-x)^2.$

These reduce to two linear equations in $r$ and $x,$ whose solution is $r=\dfrac{6}{7}$ (and $x=\dfrac{9}{7}$ ).

Thus, the correct answer is B.

2017 AMC 12A Problem 16

Solution:

Problem 16 in Other Years