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2017 AMC 12A Problem 16

Problem 16 of 25IntermediateGeometry

In the figure below, semicircles with centers at AA and BB and with radii 22 and 1,1, respectively, are drawn in the interior of, and sharing bases with, a semicircle with diameter JK‾.\overline{JK}. The two smaller semicircles are externally tangent to each other and internally tangent to the largest semicircle. A circle centered at PP 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?P?

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Solution

The large semicircle has radius 33 and center C,C, the midpoint of JK‾.\overline{JK}. Placing JJ at the origin, A=2,A=2, B=5,B=5, C=3,C=3, K=6K=6 along the base. Let rr be the radius of the circle at P.P. By tangency, PA=2+r,PA=2+r, PB=1+r,PB=1+r, and PC=3−r.PC=3-r. Dropping a perpendicular from PP to the base at horizontal position 3+x3+x with height h,h, the Pythagorean theorem gives h2=(2+r)2−(1+x)2=(3−r)2−x2=(1+r)2−(2−x)2. \begin{aligned} h^2 &=(2+r)^2-(1+x)^2 \\ &=(3-r)^2-x^2 \\ &=(1+r)^2-(2-x)^2. \end{aligned} Equating the first expression with the middle one gives 5r−x=3,5r-x=3, while equating the last expression with the middle one gives 2r+x=3.2r+x=3. Adding yields 7r=6,7r=6, so r=67.r=\dfrac{6}{7}. Thus, the correct answer is B.

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Concepts: tangent circles · Pythagorean Theorem · coordinate geometry

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