题目
题型:不详难度:来源:
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![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085852-36023.gif)
(Ⅰ)求证:AE//面PBC.
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![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085852-35921.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085852-37301.jpg)
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![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085852-86621.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085853-78957.gif)
所以EF//AB,且EF=AB,所以ABFE为平行四边形 …2分
所以AE//BF, 因为AE
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085853-40145.gif)
(Ⅱ)建立如图所示的空间直角坐标系,
则 A、B、C、D、P、E的坐标分别为A(0,0,0),
B(1,0,0),C(2,1,0),D(0,1,0),
(0,0,3),E(0,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085853-55587.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085853-65791.gif)
从而
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085854-59660.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085854-97026.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085854-91460.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085854-59660.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085854-97026.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085854-85287.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085855-33031.gif)
∴AC与PB所成角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085855-67366.gif)
(Ⅲ)法1:由于N点在面PAB内,故可设N点坐标为(x,0,z),
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085855-91346.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085855-38913.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085856-19515.gif)
化简得
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085856-31384.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085856-90208.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085853-65791.gif)
所以在面PAB内存在点N(
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085856-90208.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085853-65791.gif)
(Ⅲ)法2:在面ABCD内过D作AC的垂线交AB于G,连PG,
设N为PG的中点,连NE,则NE//DG, …10分
∵DG⊥AC,DG⊥PA,∴DG⊥面PAC 从而NE⊥面PAC …14分
解析
核心考点
试题【(本小题满分14分)_ _ .如图,ABCD是梯形,AB//CD,,PA⊥面ABCD, 且AB=1,AD=1,CD=2,PA=3,E为PD的中点】;主要考察你对空间几何体的视图与直观图等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085848-54111.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085849-62825.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085839-72654.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085839-65897.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085840-88095.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085839-65897.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085840-98824.gif)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085840-70228.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085840-57012.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085841-22875.gif)
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085841-76229.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085841-99527.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085841-98918.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085841-13141.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085827-21800.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085828-41226.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085828-20320.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085828-52696.gif)
(2)在侧面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085828-12141.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085829-64469.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085829-76010.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085829-97758.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085829-10147.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085823-31125.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085823-31327.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021085820-92419.jpg)
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