题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034649-96236.png)
PD=1,PC=
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034649-69535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034649-64835.jpg)
(Ⅰ)求证:PD⊥面ABCD;
(Ⅱ)求二面角A-PB-D的大小.
答案
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034649-41824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034650-44722.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034650-45089.png)
(II)解本小题关键是作出二面角的平面角,过O作OE⊥PB于点E,连结AE,
证明
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034650-19788.png)
(Ⅰ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034651-97998.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034651-89571.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034650-45089.png)
∴ PD⊥面ABCD………6分
(Ⅱ)解:连结BD,设BD交AC于点O,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034651-94436.png)
过O作OE⊥PB于点E,连结AE,
∵PD⊥面ABCD, ∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034651-77962.png)
又∵AO⊥BD,∴AO⊥面PDB.
∴AO⊥PB,
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034652-65070.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034652-88373.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034652-26177.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034650-19788.png)
∵ PD⊥面ABCD, ∴PD⊥BD,
∴在Rt△PDB中,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034652-66579.png)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034653-99385.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034653-86803.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034653-54264.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034654-92154.png)
故二面角A-PB-D的大小为60°.…………………14分
(也可用向量解)
核心考点
试题【(本小题满分14分)如图,四棱锥P-ABCD是底面边长为1的正方形,PD⊥BC,PD=1,PC=.PD=1,PC=,PD⊥BC。(Ⅰ)求证:PD⊥面ABCD;(】;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
如图,在三棱锥P—ABC中,PC⊥底面ABC,AB⊥BC,D,E分别是AB,PB的中点.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034645-64551.png)
(1)求证:DE∥平面PAC
(2)求证:AB⊥PB
在
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034634-23545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034635-69592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034635-69719.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034635-24034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034635-29654.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034636-19775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034636-87759.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034636-84062.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034636-24851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034636-41690.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034637-87591.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034637-62568.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034626-73714.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034626-69872.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034626-12262.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034626-25850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034626-69872.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034627-97890.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034627-66159.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034627-84604.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034628-65504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034628-82833.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034628-74894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034628-94452.png)
(2)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034628-94452.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034629-76648.png)
(3)求
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034628-74894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034629-55043.png)
A.若m∥n,m^a,则n^a | B.若m^a,mÌb,则a^b |
C.若m^a,a∥b,则m^b | D.若a^b,mÌa,则m^b |
如图,四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034612-39677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034612-85323.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034612-39814.jpg)
(Ⅰ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034613-20867.png)
(Ⅱ)当
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106034613-40411.png)
面PDB所成的角的大小。
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