扇形OAB的弦AB=18,半径为6的圆C恰与OA、OB和弧AB相切,圆D又与圆C、OA和OB相切,则圆D的半径为准确率准确率

Question

扇形OAB的弦AB=18,半径为6的圆C恰与OA、OB和弧AB相切,圆D又与圆C、OA和OB相切,则圆D的半径为__________
准确率 准确率

Answer 1

圆C半径R=6
做CE⊥OB于E,DF⊥OB与F
CE=R=6,DF=r
OC=OB-R=OB-6
OD=OB-2R-r=OB-12-r
DF/OD=CE/OC=1/2AB/OB
r/(OB-12-r) = 6/(OB-6) = 1/2*18/OB
由6/(OB-6) = 1/2*18/OB=9/OB
9OB-9*6=6OB
OB=(9*6)/(9-6)=18
带入r/(OB-12-r) = 6/(OB-6)
r/(18-12-r) = 6/(18-6)
r/(6-r) = 1/2
6-r = 2r
3r=6
r=2
圆D的半径为2