一到数学立几如 图所示,已知斜三棱柱ABC-A1B1C1的侧面A1ACC1与底面ABC垂直,∠ABC=90°,BC=2,AC=2 ,且AA1⊥A1C,AA1=A1C.(1)求侧棱A1A与底面ABC所成角的大小; (2)求侧面A1ABB1与底面ABC所成二面角的大小
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![一到数学立几如 图所示,已知斜三棱柱ABC-A1B1C1的侧面A1ACC1与底面ABC垂直,∠ABC=90°,BC=2,AC=2 ,且AA1⊥A1C,AA1=A1C.(1)求侧棱A1A与底面ABC所成角的大小; (2)求侧面A1ABB1与底面ABC所成二面角的大小](/uploads/image/z/11499825-57-5.jpg?t=%E4%B8%80%E5%88%B0%E6%95%B0%E5%AD%A6%E7%AB%8B%E5%87%A0%E5%A6%82+%E5%9B%BE%E6%89%80%E7%A4%BA%2C%E5%B7%B2%E7%9F%A5%E6%96%9C%E4%B8%89%E6%A3%B1%E6%9F%B1ABC-A1B1C1%E7%9A%84%E4%BE%A7%E9%9D%A2A1ACC1%E4%B8%8E%E5%BA%95%E9%9D%A2ABC%E5%9E%82%E7%9B%B4%2C%E2%88%A0ABC%3D90%C2%B0%2CBC%3D2%2CAC%3D2+%2C%E4%B8%94AA1%E2%8A%A5A1C%2CAA1%3DA1C.%EF%BC%881%EF%BC%89%E6%B1%82%E4%BE%A7%E6%A3%B1A1A%E4%B8%8E%E5%BA%95%E9%9D%A2ABC%E6%89%80%E6%88%90%E8%A7%92%E7%9A%84%E5%A4%A7%E5%B0%8F%EF%BC%9B+%EF%BC%882%EF%BC%89%E6%B1%82%E4%BE%A7%E9%9D%A2A1ABB1%E4%B8%8E%E5%BA%95%E9%9D%A2ABC%E6%89%80%E6%88%90%E4%BA%8C%E9%9D%A2%E8%A7%92%E7%9A%84%E5%A4%A7%E5%B0%8F)
一到数学立几如 图所示,已知斜三棱柱ABC-A1B1C1的侧面A1ACC1与底面ABC垂直,∠ABC=90°,BC=2,AC=2 ,且AA1⊥A1C,AA1=A1C.(1)求侧棱A1A与底面ABC所成角的大小; (2)求侧面A1ABB1与底面ABC所成二面角的大小
一到数学立几
如 图所示,已知斜三棱柱ABC-A1B1C1的侧面A1ACC1与底面ABC垂直,∠ABC=90°,BC=2,AC=2 ,且AA1⊥A1C,AA1=A1C.
(1)求侧棱A1A与底面ABC所成角的大小;
(2)求侧面A1ABB1与底面ABC所成二面角的大小;
(3)求顶点C到侧面A1ABB1的距离.
一到数学立几如 图所示,已知斜三棱柱ABC-A1B1C1的侧面A1ACC1与底面ABC垂直,∠ABC=90°,BC=2,AC=2 ,且AA1⊥A1C,AA1=A1C.(1)求侧棱A1A与底面ABC所成角的大小; (2)求侧面A1ABB1与底面ABC所成二面角的大小
(Ⅰ)作A1D⊥AC,垂足为D,
由面A1ACC1⊥面ABC,得A1D⊥面ABC, ∴∠A1AD为A1A与面ABC所成的角.
∵AA1⊥A1C,AA1=A1C, ∴∠A1AD=45°为所求.
(Ⅱ)作DE⊥AB,垂足为E,连A1E,则由A1D⊥面ABC,得A1E⊥AB.
∴∠A1ED是面A1ABB1与面ABC所成二面角的平面角.
由已知,AB⊥BC,得ED‖BC.又D是AC的中点,
BC=2,AC=2根号3, ∴DE=1,AD=A1D=根号3,
tgA1ED=A1D/DE=根号3. 故∠A1ED=60°为所求.
(Ⅲ)由点C作平面A1ABB1的垂线,垂足为H,则CH的长是C到平面A1ABB1的距离.
连结HB,由于AB⊥BC,得AB⊥HB. 又A1E⊥AB,
知HB‖A1E,且BC‖ED, ∴∠HBC=∠A1ED=60°.
∴CH=BCsin60°=根号3为所求.