Problems(1)
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]<spanclass=′latex−bold′>(A)</span> 42<spanclass=′latex−bold′>(B)</span> 21<spanclass=′latex−bold′>(C)</span> π2<spanclass=′latex−bold′>(D)</span> 32<spanclass=′latex−bold′>(E)</span> 22 ratiogeometry