如图,已知斜三棱柱ABC—A1B1C1中,∠BCA=90°,AC=BC=2,点A1在底面ABC上的射影恰为AC的中点D,BA1⊥AC1.1)求证:BC⊥平面A1ACC1;(2)求二面角B-AA1-C的正切值.

问题描述:

如图,已知斜三棱柱ABC—A1B1C1中,∠BCA=90°,AC=BC=2,点A1在底面ABC上的射影恰为AC的中点D,BA1⊥AC1.
1)求证:BC⊥平面A1ACC1;
(2)求二面角B-AA1-C的正切值.

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(1)A1D⊥面ABC所以A1D⊥BC又BC⊥ACAC∩A1D=D所以BC⊥面A1ACC1(2)由(1)知:BC⊥AC1,又A1B⊥AC1,A1B∩BC=B所以AC1⊥面A1BC,所以AC1⊥A1C所以ACC1A1是菱形因为D是AC中点,所以∠A1AD=60°,作CN⊥AA1于点N,连结BN...