题目内容
已知三棱柱ABC—A1B1C1中底面边长和侧棱长均为a,侧面A1ACC1⊥底面ABC,A1B=
(1)求异面直线AC与BC1所成角的余弦值;
(2)求证:A1B⊥面AB1C.
解析:如图过点B作BO⊥AC,垂足为点O,则BO⊥侧面ACC1A1,连结A1O.
在Rt△A1BO中,
A1B=,BO=
,
∴A1O=,又AA1=a,AO=
,
∴△A1AO为直角三角形.
∴A1O⊥AC,A1O⊥底面ABC.
(1)∵A1C1∥AC,
∴∠BC1A1或其补角为异面直线AC与BC1所成的角.
∵A1O面ABC,AC⊥BO,
∴AC⊥A1B,∴A1C1⊥A1B.
在Rt△A1BC1中,A1B=a,A1C1=a,
∴BC1=,∴cos∠BC1A1=
.
∴异面直线AC与BC1所成角的余弦值为.
(2)∵四边形ABB1A1为菱形,∴AB1⊥A1B.
又A1B⊥AC,
∴A1B⊥面AB1C.

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