如图,梯形ABCD中,AD∥BC,AC,BD相交于点O,过O作BC的平行线分别交AB,CD于点E,F. (1)求证:OE=OF; (2)若AD=3,BC=4,求EF的长.
问题描述:
如图,梯形ABCD中,AD∥BC,AC,BD相交于点O,过O作BC的平行线分别交AB,CD于点E,F.
(1)求证:OE=OF;
(2)若AD=3,BC=4,求EF的长.
答
(1)证明:∵AD∥BC,∴△AOE∽△ABC,△DOF∽△DBC,∴OEBC=AOAC,OFBC=DFDC,又∵由AD∥BC得,△ACD∽△OCF,∴AOAC=DFDC,∴OEBC=OFBC,∴OE=OF;(2)∵AD∥BC,∴△AOD∽△BOC,∴AOOC=ADBC=34,∴AOAC=33+4=3...