如图,已知三棱柱ABC-A1B1C1中,侧棱A A1⊥底面ABC,AB⊥BC; (Ⅰ)求证:平面A1BC⊥侧面A1ABB1. (Ⅱ)若AA1=AC=a,直线AC与平面A1BC所成的角为π6,求AB的长.

问题描述:

如图,已知三棱柱ABC-A1B1C1中,侧棱A A1⊥底面ABC,AB⊥BC;

(Ⅰ)求证:平面A1BC⊥侧面A1ABB1
(Ⅱ)若AA1=AC=a,直线AC与平面A1BC所成的角为

π
6
,求AB的长.

(Ⅰ)证明:如图,已知AA1⊥平面ABC,BC⊂面ABC,∴AA1⊥BC,又已知AB⊥BC,且AB∩AA1=A,∴BC⊥平面AA1BB1,而BC⊂面A1BC,∴平面A1BC⊥面A1ABB1;(Ⅱ)过点A在平面AA1BB1内作AD⊥A1B,垂足是D,连结CD,∵平面A1...