题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012915-73640.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-82800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-23499.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-35381.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-42863.png)
(Ⅱ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-75398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-48141.jpg)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-97434.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-44567.png)
又∵EA
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-97434.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-44567.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-97434.jpg)
∴四边形AFPE是平行四边形,∴AF∥EP,
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-13907.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012917-10251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-42863.png)
∴AF∥面BDE.…………………………………………4分
(Ⅱ)以CA、CD所在直线分别作为x轴,z轴,以过C点和AB平行的直线作为y轴,建立如图所示坐标系.…………………5分
由
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-23499.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012918-37998.png)
∵面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012918-59046.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012918-86191.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012918-68554.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012919-12086.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012919-84444.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012919-70984.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012919-32227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012920-35794.png)
设面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-42863.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012920-61468.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012920-66633.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012920-94993.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012920-24691.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012921-56582.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012921-55331.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012921-33791.png)
所以n=(1,1,2)是面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012920-35794.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012921-30468.png)
图形可知二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012916-75398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012922-18950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012922-14816.png)
解析
核心考点
举一反三
(1)求直线C1B与底面ABC所成角的正切值;
(2)在棱CC1(不包括端点C、C1)上确定一点E的位置,使EA⊥EB1(要求说明理由);
(3)在(2)的条件下,若AB=,求二面角A-EB1-A1的大小.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012912-41430.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012903-22901.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012903-72489.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012904-37947.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012904-86374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012904-95133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012904-40494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012904-77475.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012905-80026.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012904-40494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012905-41548.png)
(I)求证:CN//平面 AMB1;
(II)若二面角A-MB1-C为45°,求CC1的长.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012900-98189.jpg)
(1)求证:B1G⊥CF;
(2)求二面角F-EC-D的余弦值。
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012853-29753.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012843-93911.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012843-92450.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022012843-11204.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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