题目内容
如图, 已知四边形ABCD和BCEG均为直角梯形,AD∥BC,CE∥BG,且
,平面ABCD⊥平面BCEG,BC=CD=CE=2AD=2BG=2.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240421347613434.jpg)
(1)求证: EC⊥CD;
(2)求证:AG∥平面BDE;
(3)求:几何体EG-ABCD的体积.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134761790.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240421347613434.jpg)
(1)求证: EC⊥CD;
(2)求证:AG∥平面BDE;
(3)求:几何体EG-ABCD的体积.
(1)证明过程详见解析;(2)证明过程详见解析;(3)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134777375.png)
试题分析:(1)要证
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134792550.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134808440.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134823526.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134839551.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134839549.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134855543.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134808440.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134823526.png)
(2)过G作GN⊥CE交BE于M,连 DM,由题设可证四边形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134901527.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134901618.png)
从而由直线与平面平行的判定定理,可证AG∥平面BDE;
(3)欲求几何体EG-ABCD的体积,可先将该几何体分成一个四棱锥
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042134917636.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135011575.png)
试题解析:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240421350264936.jpg)
(1)证明:由平面ABCD⊥平面BCEG,
平面ABCD∩平面BCEG=BC,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135042601.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135057195.png)
又CD
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135057214.png)
(2)证明:在平面BCDG中,过G作GN⊥CE交BE于M,连DM,则由已知知;MG=MN,MN∥BC∥DA,且
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135089846.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135057195.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135057195.png)
∵DM
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135120217.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135135273.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135057195.png)
(3)解:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240421351671568.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824042135182927.png)
![](http://thumb.zyjl.cn/images/loading.gif)
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