题目内容
如图,已知正方体
,
分别为各个面的对角线;
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240133213725375.jpg)
(1)求证:
;
(2)求异面直线
所成的角.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321340787.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321356811.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240133213725375.jpg)
(1)求证:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321403788.png)
(2)求异面直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321418604.png)
(1)∵
∴
∴
,又∵
∴
(2)![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321543397.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240133214341222.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321465592.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321496654.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240133215121505.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321403788.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321543397.png)
试题分析:(1)在正方体
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321340787.png)
∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240133215901228.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321465592.png)
又∵四边形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321621608.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321496654.png)
又∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240133216521506.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321403788.png)
(2)∵
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321699562.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321715524.png)
∴四边形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321746480.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321762574.png)
∴
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321793542.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321418604.png)
连接
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013322074442.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013322105554.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013322120712.png)
∴异面直线
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321418604.png)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824013321543397.png)
点评:要证线面垂直需证直线垂直于平面内两条相交直线,求异面直线所成角的步骤:空间取一点,过该点作两异面直线的平行线,找到异面直线所成角,求解三角形得到所求角
![](http://thumb.zyjl.cn/images/loading.gif)
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