题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-80414.jpg)
本题满分15分)如图,在矩形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-27246.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-31113.gif)
在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-62590.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-70662.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-72186.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-28063.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-13818.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-92043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-33975.gif)
(Ⅰ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-46083.gif)
(Ⅱ)点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-16267.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-77718.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-71889.gif)
边形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-22102.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-60043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-48778.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-30822.gif)
的长。
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-85073.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-62583.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-51234.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-86315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-25272.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-33186.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-33021.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031801-67358.jpg)
又因为平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031801-94194.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031801-47082.gif)
如图建立空间直角坐标系A-xyz
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031801-99961.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031801-65072.gif)
F(4,0,0),D(10,0,0).
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031802-45857.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031802-80375.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031802-82319.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031802-66970.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031802-42966.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-52981.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031802-80375.gif)
所以
6x=0.
取
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-39157.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-61030.gif)
又平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-56799.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-31200.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031804-85005.gif)
所以二面角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-85073.gif)
(Ⅱ)解:设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031804-92883.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031804-37468.gif)
因为翻折后,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-60043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031804-62724.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-64772.gif)
故,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-49570.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-22767.gif)
经检验,此时点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-13550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-70392.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-62583.gif)
方法二:
(Ⅰ)解:取线段
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-72186.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-97841.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-66134.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-21626.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-84157.gif)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-81407.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031807-44812.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-97841.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-72186.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031807-17178.gif)
又因为平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031807-48765.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031807-39007.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-56799.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-54260.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-56799.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-48727.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031803-56799.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-54260.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-66134.gif)
又因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-21626.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-97841.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-66134.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-72186.gif)
易知
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-94079.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-90118.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-94079.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031807-39007.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-66134.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031806-66134.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031807-39007.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-29103.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031809-76398.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031809-57088.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031809-62070.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031809-29380.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031801-65072.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031808-94079.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031809-86246.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031810-23101.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031810-76179.gif)
故二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031810-17446.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-85073.gif)
(Ⅱ)解:设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031810-98106.gif)
因为翻折后,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-60043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031811-69062.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-64772.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031811-44915.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031811-60831.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031811-33835.gif)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-22767.gif)
经检验,此时点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-13550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031805-70392.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-62583.gif)
核心考点
试题【本题满分15分)如图,在矩形中,点分别在线段上,.沿直线将翻折成,使平面. (Ⅰ)求二面角的余弦值;(Ⅱ)点分别在线段上,若沿直线将四边形向上翻折,使与重合,求】;主要考察你对空间几何体的结构特征等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031750-99825.gif)
(Ⅰ)求证:FH∥平面EDB;
(Ⅱ)求证:AC⊥平面EDB;
(Ⅲ)求四面体B—DEF的体积;
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-83704.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-41305.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-48766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-41305.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-48766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031745-90624.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031745-65656.gif)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-42586.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-25305.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-25305.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-71098.jpg)
(Ⅰ)证明:DE为异面直线AB
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-25305.gif)
(Ⅱ)设异面直线AB
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031741-63425.gif)
如图,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031731-98902.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031731-93253.gif)
平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031731-90004.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-71453.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-46102.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-71453.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-76034.gif)
(1)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-72468.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-11871.gif)
(2)求平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-19839.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031732-71453.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031722-62491.gif)
(Ⅰ)求异面直线A1M和C1D1所成的角的正切值;
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031722-44142.jpg)
(Ⅱ)证明:平面ABM⊥平面A1B1M1
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