如图,BD为⊙O的直径,AB=AC,AD交BC于E,AE=2,ED=4. (1)求证:△ABE∽△ADB. (2)求AB长.

问题描述:

如图,BD为⊙O的直径,AB=AC,AD交BC于E,AE=2,ED=4.

(1)求证:△ABE∽△ADB.
(2)求AB长.

(1)证明:∵AB=AC,
∴∠ABC=∠C,
∵∠C=∠D,
∴∠ABC=∠D,(2分)
又∵∠BAE=∠DAB,
∴△ABE∽△ADB;
(2)∵△ABE∽△ADB,

AB
AD
=
AE
AB

∴AB2=AD•AE=(AE+ED)•AE=(2+4)×2=12,(2分)
∴AB=2
3
.(2分)