题目内容
(本小题满分12分)如图,在底面为直角梯形的四棱锥![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329322858.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329338706.png)
,
,BC=6.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503294165178.png)
(Ⅰ)求证:![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329432724.png)
(Ⅱ)求二面角
的大小.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329322858.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329338706.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329354488.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329385899.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503294165178.png)
(Ⅰ)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329432724.png)
(Ⅱ)求二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329494499.png)
(Ⅰ) 证明见解析
(Ⅱ)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329541806.png)
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329541806.png)
解法一:(Ⅰ)
平面
,
平面
.
.
又
,
.
,
,
,即
.
又
.
平面
.
(Ⅱ)过
作
,垂足为
,连接
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503301183782.png)
平面
,
是
在平面
上的射影,由三垂线定理知
,
为二面角
的平面角.
又
,
,
,
又
,
,
.
由
得
.
在
中,
,
.
二面角
的大小为
.
解法二:(Ⅰ)如图,建立坐标系,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503309144311.png)
则
,
,
,
,
,
,
,
,
,
.
,
,
又
,
平面
.
(Ⅱ)设平面
的法向量为
,
则
,
,
又
,
,
解得![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331569984.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503315841073.png)
平面
的法向量取为
,
,
.
二面角
的大小为
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329556432.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329572534.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329634418.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329572534.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329681564.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329697994.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503298061006.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329822676.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329853663.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329868692.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329884580.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329900625.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329915421.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329931465.png)
(Ⅱ)过
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329993322.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330024586.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330056303.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330102405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503301183782.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330134472.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329931465.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330180390.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330102405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329931465.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330258582.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330274543.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330290572.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330321881.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330336850.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330383891.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330399619.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330414637.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330430490.png)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330446871.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503305241141.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330602605.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503307261160.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503307581095.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330804202.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330290572.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330851774.png)
解法二:(Ⅰ)如图,建立坐标系,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503309144311.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330929537.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330945699.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330960749.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330976596.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331007562.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331023705.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331038853.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331038822.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331070676.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331085671.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331101572.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329884580.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329900625.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329915421.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329931465.png)
(Ⅱ)设平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331226460.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331272642.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331288629.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331413611.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331475876.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331506737.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503315381154.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331569984.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503315841073.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329931465.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503316161081.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150331694500.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503317091077.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330804202.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150330290572.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150329541806.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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