如图,三棱柱ABC-A1B1C1中,侧棱AA1⊥底面ABC,△ABC是边长为2的正三角形,点P在A1B上,且AB⊥CP.证明 1.P为A1B中点.2.若A1B⊥AC1,求三棱锥P-A1AC的体积.
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![如图,三棱柱ABC-A1B1C1中,侧棱AA1⊥底面ABC,△ABC是边长为2的正三角形,点P在A1B上,且AB⊥CP.证明 1.P为A1B中点.2.若A1B⊥AC1,求三棱锥P-A1AC的体积.](/uploads/image/z/9491725-37-5.jpg?t=%E5%A6%82%E5%9B%BE%2C%E4%B8%89%E6%A3%B1%E6%9F%B1ABC-A1B1C1%E4%B8%AD%2C%E4%BE%A7%E6%A3%B1AA1%E2%8A%A5%E5%BA%95%E9%9D%A2ABC%2C%E2%96%B3ABC%E6%98%AF%E8%BE%B9%E9%95%BF%E4%B8%BA2%E7%9A%84%E6%AD%A3%E4%B8%89%E8%A7%92%E5%BD%A2%2C%E7%82%B9P%E5%9C%A8A1B%E4%B8%8A%2C%E4%B8%94AB%E2%8A%A5CP.%E8%AF%81%E6%98%8E+1.P%E4%B8%BAA1B%E4%B8%AD%E7%82%B9.2.%E8%8B%A5A1B%E2%8A%A5AC1%2C%E6%B1%82%E4%B8%89%E6%A3%B1%E9%94%A5P-A1AC%E7%9A%84%E4%BD%93%E7%A7%AF.)
如图,三棱柱ABC-A1B1C1中,侧棱AA1⊥底面ABC,△ABC是边长为2的正三角形,点P在A1B上,且AB⊥CP.证明 1.P为A1B中点.2.若A1B⊥AC1,求三棱锥P-A1AC的体积.
如图,三棱柱ABC-A1B1C1中,侧棱AA1⊥底面ABC,△ABC是边长为2的正三角形,点P在A1B上,
且AB⊥CP.
证明 1.P为A1B中点.
2.若A1B⊥AC1,求三棱锥P-A1AC的体积.
如图,三棱柱ABC-A1B1C1中,侧棱AA1⊥底面ABC,△ABC是边长为2的正三角形,点P在A1B上,且AB⊥CP.证明 1.P为A1B中点.2.若A1B⊥AC1,求三棱锥P-A1AC的体积.
1.取AB中点M,连接CM,PM
AB⊥CP
AB⊥CM
AB⊥平面CPM
AB⊥PM
PM//AA1,M为AB中点 所以P为A1B中点
2.取A1B1中点N,连接AN,C1N
A1B⊥AC1
C1N⊥A1B
A1B⊥平面AC1N
A1B⊥AN △NA1A与△A1AB相似
AB=2 AA1=√2
点P到平面A1AC的距离d等于点B到平面A1AC的距离的一半
d=√3
SA1AC=1/2*2*√2=√2
VP-A1AC=1/3*d*S=√6/3
1. 取AB中点M,连接CM,PM
AB⊥CP
AB⊥CM
AB⊥平面CPM
AB⊥PM
PM//AA1, M为AB中点 所以P为A1B中点
2. 取A1B1中点N,连接AN,C1N
A1B⊥AC1
C1N⊥A1B
A1B⊥平面AC1N
A1B⊥AN △NA1A与△A1AB相似
AB=2 AA1=√2
...
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1. 取AB中点M,连接CM,PM
AB⊥CP
AB⊥CM
AB⊥平面CPM
AB⊥PM
PM//AA1, M为AB中点 所以P为A1B中点
2. 取A1B1中点N,连接AN,C1N
A1B⊥AC1
C1N⊥A1B
A1B⊥平面AC1N
A1B⊥AN △NA1A与△A1AB相似
AB=2 AA1=√2
点P到平面A1AC的距离d等于点B到平面A1AC的距离的一半
d=√3
SA1AC=1/2*2*√2=√2
VP-A1AC=1/3*d*S=√6/3
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