题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031559-53845.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031600-42969.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031600-84665.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031600-15575.png)
(Ⅰ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031600-44869.png)
(Ⅱ)若E为PB的中点时,求AE与平面PDB所成的角的大小.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031601-96533.jpg)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031601-45775.png)
解析
(Ⅰ)欲证平面AEC⊥平面PDB,根据面面垂直的判定定理可知在平面AEC内一直线与平面PDB垂直,而根据题意可得AC⊥平面PDB;
(Ⅱ)设AC∩BD=O,连接OE,根据线面所成角的定义可知∠AEO为AE与平面PDB所的角,在Rt△AOE中求出此角即可.
解:以D为原点建立空间直角坐标系
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031601-31025.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031602-15501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031602-88817.png)
(Ⅰ)∵,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031602-19104.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031602-64996.png)
∴AC⊥DP,AC⊥DB,∴AC⊥平面PDB,
∴平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031600-44869.png)
(Ⅱ)当E为PB的中点时,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031603-81128.png)
设AC∩BD=O,连接OE,
由(Ⅰ)知AC⊥平面PDB于O,
∴∠AEO为AE与平面PDB所的角,
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031603-65641.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031603-94214.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031603-32320.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031601-45775.png)
核心考点
试题【(本小题满分12分)四棱锥的底面是正方形,,点E在棱PB上.若AB=,(Ⅰ)求证:平面; (Ⅱ)若E为PB的中点时,求AE与平面PDB所成的角的大小.】;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
A.相交 | B.平行 | C.异面 | D.以上都有可能 |
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031550-30077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031550-47016.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031550-30077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031550-47016.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031550-30077.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031550-67794.jpg)
(1)求证:P、C、D、Q四点共面;
(2)求证:QD⊥AB.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031540-78386.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031540-51891.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031541-93694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031541-16235.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031541-10248.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031541-35675.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031542-93283.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031542-83894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031542-87429.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031542-79541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031543-52361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031543-66669.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031542-87429.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031543-79139.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031533-36685.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031533-76554.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031533-76547.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031533-65323.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031533-98844.png)
(2)若直线
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031534-10526.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031533-36685.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031534-49587.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031534-46485.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031534-69333.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031522-57395.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031522-22579.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031523-80757.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031523-65290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031523-62990.png)
给出下列四个命题:
①若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031523-43276.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031524-20650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031524-76062.png)
②若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031524-41937.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031524-23680.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031525-58579.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031525-32230.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031525-55804.png)
③若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031525-39408.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031526-12255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031526-57456.png)
④若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031523-65290.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031523-62990.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031526-87416.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031525-32230.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031526-57348.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031527-52468.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106031527-78211.png)
其中真命题的序号是( )
A.①③④ | B.①②③ | C.①③ | D.②④ |
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