MathDB

Semicircles

POQ

and

ROS

pass through the center of circle

O

. What is the ratio of the combined areas of the two semicircles to the area of circle

O

? [asy] import graph; size(7.5cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-6.27,xmax=10.01,ymin=-5.65,ymax=10.98; draw(circle((0,0),2)); draw((-3,0)--(3,0),EndArrow(6)); draw((0,-3)--(0,3),EndArrow(6)); draw(shift((0.01,1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,179.76,359.76)); draw(shift((-0.01,-1.42))*xscale(1.41)*yscale(1.41)*arc((0,0),1,-0.38,179.62)); draw((-1.4,1.43)--(1.41,1.41)); draw((-1.42,-1.41)--(1.4,-1.42)); label("

P(-1,1)

",(-2.57,2.17),SE*lsf); label("

Q(1,1)

",(1.55,2.21),SE*lsf); label("

R(-1,-1)

",(-2.72,-1.45),SE*lsf); label("

S(1,-1)

",(1.59,-1.49),SE*lsf); dot((0,0),ds); label("

O

",(-0.24,-0.35),NE*lsf); dot((1.41,1.41),ds); dot((-1.4,1.43),ds); dot((1.4,-1.42),ds); dot((-1.42,-1.41),ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

<span class='latex-bold'>(A)</span>\ \frac{\sqrt 2}4 \qquad<span class='latex-bold'>(B)</span>\ \frac 12 \qquad<span class='latex-bold'>(C)</span>\ \frac{2}{\pi} \qquad<span class='latex-bold'>(D)</span>\ \frac{2}{3}\qquad<span class='latex-bold'>(E)</span>\ \frac{\sqrt 2}{2}

Problem Statement