Measure of Angle
Source:
February 11, 2009
rotation
Problem Statement
In the figure, is tangent to semicircle ; is tangent to semicircle ; is a straight line; the arcs are indicated in the figure. Angle is measured by:
[asy]unitsize(1.2cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;pair O1=(0,0), O2=(3,0), Sp=(-2,0), R=(2,0), T=(4,0);
pair A=O1+2*dir(60), B=O2+dir(85);
pair Pa=rotate(90,A)*O1, Pb=rotate(-90,B)*O2;
pair P=extension(A,Pa,B,Pb);
pair[] dots={Sp,R,T,A,B,P};draw(P--P+5*(A-P));
draw(P--P+5*(B-P));
clip((-2,0)--(-2,2.5)--(4,2.5)--(4,0)--cycle);
draw(Arc(O1,2,0,180)--cycle);
draw(Arc(O2,1,0,180)--cycle);dot(dots);
label("",Sp,S);
label("",R,S);
label("",T,S);
label("",A,NE);
label("",B,N);
label("",P,NNE);
label("",midpoint(Arc(O1,2,0,60)),SW);
label("",midpoint(Arc(O2,1,85,180)),SE);
label("",midpoint(Arc(O1,2,60,180)),SE);
label("",midpoint(Arc(O2,1,0,85)),SW);[/asy] (A)\ \frac {1}{2}(a \minus{} b) \qquad (B)\ \frac {1}{2}(a \plus{} b) \qquad (C)\ (c \minus{} a) \minus{} (d \minus{} b) \qquad (D)\ a \minus{} b \qquad (E)\ a \plus{} b