Problem

2013 AMC 8 Problem 23

Angle ABC of \triangle ABC is a right angle. The sides of \triangle ABC are the diameters of semicircles as shown. The area of the semicircle on \overline{AB} equals 8\pi, and the arc of the semicircle on \overline{AC} has length 8.5\pi. What is the radius of the semicircle on \overline{BC}?

[asy] import graph; pair A,B,C; A=(0,8); B=(0,0); C=(15,0); draw((0,8)..(-4,4)..(0,0)--(0,8)); draw((0,0)..(7.5,-7.5)..(15,0)--(0,0)); real theta = aTan(8/15); draw(arc((15/2,4),17/2,-theta,180-theta)); draw((0,8)--(15,0)); dot(A); dot(B); dot(C); label("$A$", A, NW); label("$B$", B, SW); label("$C$", C, SE);[/asy]

\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9

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


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Category: AMC 8
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