如图,已知AB是⊙O的直径,直线CD与⊙O相切于点C,过A作AD⊥CD,D为垂足. (1)求证:AC平分∠DAB. (2)若AD=3,AC=15,求AB的长.

问题描述:

如图,已知AB是⊙O的直径,直线CD与⊙O相切于点C,过A作AD⊥CD,D为垂足.

(1)求证:AC平分∠DAB.
(2)若AD=3,AC=

15
,求AB的长.

(1)证明:连接OC,∵直线CD与⊙O相切于点C,∴OC⊥CD,∵AD⊥CD,∴OC∥AD,∴∠OCA=∠DAC,∵OC=OA,∴∠OCA=∠OAC,∴∠DAC=∠OAC,∴AC平分∠DAB;(2)连接BC,∴∠DAC=∠OAC,∠ADC=∠ACB=90°,∴△DAC∽△C...