题目
题型:不详难度:来源:
(1)证明:AP⊥BC;
(2)在线段AP上是否存在点M,使得二面角A﹣MC﹣β为直二面角?若存在,求出AM的长;若不存在,请说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014506-14646.png)
答案
解析
则O(0,0,0),A(0,﹣3,0),B(4,2,0),C(﹣4,2,0),P(0,0,4)
(1)则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014507-58213.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014507-77125.png)
由此可得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014507-58213.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014507-79495.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014507-88179.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014507-79495.png)
即AP⊥BC
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014508-35491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014508-65358.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014508-35491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014508-82967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014508-27962.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014509-55138.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014508-27962.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014509-67207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014509-56373.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014510-10482.png)
设平面BMC的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014510-85858.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014510-57955.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014510-80838.png)
令b=1,则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014510-17747.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014511-42524.png)
平面APC的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014511-97875.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014511-26799.png)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014511-29897.png)
令x=5
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014512-20719.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014512-89033.png)
得4﹣3
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014512-37089.png)
解得λ=
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014512-40985.png)
故AM=3
综上所述,存在点M符合题意,此时AM=3
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014513-60772.png)
核心考点
试题【(2011•浙江)如图,在三棱锥P﹣ABC中,AB=AC,D为BC的中点,PO⊥平面ABC,垂足O落在线段AD上,已知BC=8,PO=4,AO=3,OD=2(1】;主要考察你对向量与空间位置关系等知识点的理解。[详细]
举一反三
(1)证明:AA1⊥BD;
(2)证明:CC1∥平面A1BD.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014502-86767.png)
(1)证明:BD⊥平面PAC;
(2)若PA=1,AD=2,求二面角B﹣PC﹣A的正切值.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014449-52655.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014437-22656.png)
(1)求PA的长;
(2)求二面角B﹣AF﹣D的正弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014438-62601.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014413-62536.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014413-27237.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014413-55044.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014414-73866.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014414-75566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014414-25714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014414-74804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014415-57747.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014415-98847.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014415-29232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014415-48253.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014416-80738.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014416-73179.png)
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014416-56338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014416-23540.png)
(3)设点
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014417-78867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014417-94681.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014417-83013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014417-52733.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014418-27607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014418-47119.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014353-52853.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014353-61747.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014353-98637.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014353-97687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014354-65777.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014354-14767.png)
(1)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014354-90602.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014354-93343.png)
(2)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014355-74883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014355-82151.png)
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014355-30206.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014353-52853.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014355-17675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014356-42301.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106014356-21927.png)
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