题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-76996.png)
(Ⅰ)当E是棱CC1中点时,求证:CF∥平面AEB1;
(Ⅱ)在棱CC1上是否存在点E,使得二面角A—EB1—B的余弦值是
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-89026.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-14017.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-27503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-89026.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-75108.png)
解析
试题分析:(Ⅰ)根据直线平行平面的判定定理,需要在平面AEB1内找一条与CF平行的直线.根据题设,可取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-60327.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-91315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-60896.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-20062.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-54133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-18861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-54133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-97612.png)
试题解析:(Ⅰ)证明:取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-60327.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-91315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-81374.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-28698.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-90328.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-60327.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030327-21829.png)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030327-53993.png)
∴四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-60896.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-20062.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030327-99197.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030327-94900.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030328-52729.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030327-94900.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030328-75308.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030327-94900.png)
(Ⅱ)解:由于
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-54133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-18861.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-54133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030326-97612.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030328-97233.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030328-32641.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030328-43541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030329-78959.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030329-94180.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030329-16801.png)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030329-44946.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030330-45711.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030330-86689.png)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030330-56770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030330-22707.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030331-75715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030331-60788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030331-60788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030331-40211.png)
∵二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030331-50824.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-89026.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030331-21115.png)
解之得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030332-31527.png)
∴在棱
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-14017.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-27503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-89026.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030325-75108.png)
核心考点
试题【如图,三棱柱ABC—A1B1C1的侧棱AA1⊥底面ABC,∠ACB = 90°,E是棱CC1上动点,F是AB中点,AC = 1,BC = 2,AA1 = 4.(】;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
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-53245.png)
四个命题:①若a∥b ,a∥
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-81219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-53245.png)
②若a∥b ,b 与
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-53245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-53245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-53245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-53245.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-54106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030315-42963.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030306-85804.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030307-32984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030307-43823.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030307-46163.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030302-70257.png)
(Ⅰ)求异面直线A1B与 B1C所成角的大小;(Ⅱ)求证:平面A1BD∥平面B1CD1.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030253-80035.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030253-79453.png)
求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030253-21909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030253-25640.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030245-97151.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106030245-86462.png)
(1)求证:EF∥平面PCD;
(2)求三棱锥F﹣PCD的体积.
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