题目
题型:不详难度:来源:
已知在四棱锥P-ABCD中,底面ABCD是边长为4的正方形,△PAD是正三角形,平面PAD⊥平面ABCD,E、F、G分别是PA、PB、BC的中点.
(1)求证:EF
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083148-10916.gif)
(2)求平面EFG与平面ABCD所成锐二面角的大小;
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-89951.jpg)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-68709.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-53926.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-64605.jpg)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-93948.gif)
∵E、F为PA、PB的中点,
∴EF//AB,∴EF
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083148-10916.gif)
(II)解:过P作AD的垂线,垂足为O,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-88771.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083149-11073.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-61255.gif)
取AO中点M,连OG,,EO,EM,
∵EF //AB//OG,
∴OG即为面EFG与面ABCD的交线
又EM//OP,则EM
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-61255.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083148-10916.gif)
故OG
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083148-10916.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-27959.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-35359.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-35276.gif)
∴tan
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-27959.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-35276.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-27959.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-61905.gif)
∴平面EFG与平面ABCD所成
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083151-96913.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-61905.gif)
方法2:(I)证明:过P作P O
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083148-10916.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083151-61956.gif)
则PO
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-61255.gif)
∵PA=PD
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083151-66747.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083151-75786.gif)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083152-77065.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083152-78792.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083152-47376.gif)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083152-76696.gif)
∴EF
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083148-10916.gif)
(II)解:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083153-41247.gif)
设平面EFG的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083153-24739.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083153-31462.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083153-96964.gif)
平面ABCD的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083153-85718.gif)
平面EFG与平面ABCD所成锐二面角的余弦值是:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083154-42189.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083150-61905.gif)
解析
核心考点
试题【((本小题满分12分)已知在四棱锥P-ABCD中,底面ABCD是边长为4的正方形,△PAD是正三角形,平面PAD⊥平面ABCD,E、F、G分别是PA、PB、BC】;主要考察你对线线角等知识点的理解。[详细]
举一反三
如图7,在直三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083109-99739.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083109-63997.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083109-71081.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083109-54813.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083110-62573.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083110-25826.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083110-21720.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083110-29423.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083110-34639.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083110-24050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083102-96631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083102-22259.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083102-11042.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083102-96631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083103-61297.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083103-92651.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083102-96631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083103-34063.gif)
A.![]() | B.![]() | C.![]() | D.![]() |
如图,在三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083047-17703.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083047-69215.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083047-99489.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083048-16600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083048-15034.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083048-15369.gif)
(1)求棱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083048-38325.gif)
(2)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083048-44050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083049-16612.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083049-42518.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083043-18371.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083043-96950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083044-87112.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083044-59213.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083044-76427.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083044-76427.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022083043-18371.png)
A.与![]() | B.至多与![]() |
C.与![]() | D.至少与![]() |
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