题目内容
(12分)如图,直三棱柱
中,
,
,
为棱
的中点.
(1)求证:
平面
;
(2)求
与平面ADC所成角的正弦值.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101840503.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101855566.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101887456.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101902202.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101965250.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231501019802218.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101996274.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102136356.gif)
(2)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102152344.gif)
(1)
平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102136356.gif)
(2)
与平面ADC所成的角为
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102355795.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101996274.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102136356.gif)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102152344.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102323415.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102355795.gif)
(1)∵
,∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102386343.gif)
又∵长方形A1ABB1中
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102464343.gif)
∵
,
且![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102495370.gif)
∴
平面
……6分
(2)∵
平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102136356.gif)
∴
与平面ADC所成的角为
,……8分
设AB=1,则A1D=
,
∴
……12分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102370387.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102386343.gif)
又∵长方形A1ABB1中
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102448519.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102464343.gif)
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102386343.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102464343.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102495370.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101996274.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102136356.gif)
(2)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150101996274.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102136356.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102152344.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102323415.gif)
设AB=1,则A1D=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102635225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102651328.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150102355795.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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