如图,在梯形ABCD中,AD∥BC,AB=DC,AC⊥BD于点O,过点A作AE⊥BC于点E,若BC=2AD=8,则tan∠ABE=_.
问题描述:
如图,在梯形ABCD中,AD∥BC,AB=DC,AC⊥BD于点O,过点A作AE⊥BC于点E,若BC=2AD=8,则tan∠ABE=______.
答
过D点作DF∥AC交BC的延长线于点F,
∵AC⊥BD于点O,
∴BD⊥FD,
∵AD∥BC,
∴AD=CF,
∴BF=BC+CF=8+4=12,
∵AC=BD,
∴BD=DF,
∴AC=BD=12÷
=6
2
,
2
∴AE=
=6,
(6
)2−62
2
∴tan∠ABE=
=AE BE
=3.6 2
故答案为:3.