Chord Through Two Circles
Source:
December 30, 2006
geometryrectangletrigonometryrotationAMCAIMEinequalities
Problem Statement
In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At , one of the points of intersection, a line is drawn in such a way that the chords and have equal length. Find the square of the length of .[asy]unitsize(2.5mm);
defaultpen(linewidth(.8pt)+fontsize(12pt));
dotfactor=3;pair O1=(0,0), O2=(12,0);
path C1=Circle(O1,8), C2=Circle(O2,6);
pair P=intersectionpoints(C1,C2)[0];
path C3=Circle(P,sqrt(130));
pair Q=intersectionpoints(C3,C1)[0];
pair R=intersectionpoints(C3,C2)[1];draw(C1);
draw(C2);
//draw(O2--O1);
//dot(O1);
//dot(O2);
draw(Q--R);label("",Q,N);
label("",P,dir(80));
label("",R,E);
//label("12",waypoint(O1--O2,0.4),S);[/asy]