MathDB
2013/8/23/Semicircles

Source:

November 27, 2013
geometryPythagorean TheoremAMC

Problem Statement

Angle ABCABC of ABC\triangle ABC is a right angle. The sides of ABC\triangle ABC are the diameters of semicircles as shown. The area of the semicircle on AB\overline{AB} equals 8π8\pi, and the arc of the semicircle on AC\overline{AC} has length 8.5π8.5\pi. What is the radius of the semicircle on BC\overline{BC}? [asy] import graph; 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)); label("AA", (0,8), NW); label("BB", (0,0), SW); label("CC", (15,0), SE);[/asy]
<spanclass=latexbold>(A)</span> 7<spanclass=latexbold>(B)</span> 7.5<spanclass=latexbold>(C)</span> 8<spanclass=latexbold>(D)</span> 8.5<spanclass=latexbold>(E)</span> 9<span class='latex-bold'>(A)</span>\ 7 \qquad <span class='latex-bold'>(B)</span>\ 7.5 \qquad <span class='latex-bold'>(C)</span>\ 8 \qquad <span class='latex-bold'>(D)</span>\ 8.5 \qquad <span class='latex-bold'>(E)</span>\ 9