2006 AMC 12A Problem 13

The vertices of a 3−4−53-4-5 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles?

[asy] unitsize(5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair B=(0,0), C=(5,0); pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0]; draw(A--B--C--cycle); draw(Circle(C,3)); draw(Circle(A,1)); draw(Circle(B,2)); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("3",midpoint(B--A),NW); label("4",midpoint(A--C),NE); label("5",midpoint(B--C),S);[/asy]

(A) 12π(B) 25π2(C) 13π(D) 27π2(E) 14π\mathrm{(A) \ } 12\pi\qquad \mathrm{(B) \ } \frac{25\pi}{2}\qquad \mathrm{(C) \ } 13\pi\qquad \mathrm{(D) \ } \frac{27\pi}{2}\qquad \mathrm{(E) \ } 14\pi

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

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Category: AMC 12A
Points: 3
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