题目内容
(本小题满分14分)如下图(5),在三棱锥
中,
分别是
的中点,
,
。
(1)求证:
平面
;
(2)求异面直线
与
所成角的余弦值;
(3)求点
到平面
的距离。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947256316.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947272248.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947287325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947318617.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947350405.gif)
(1)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947365261.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947381275.gif)
(2)求异面直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947396235.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947396242.gif)
(3)求点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947412204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947428270.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231459474431585.gif)
(1)证明:连接
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947474502.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231459474902123.gif)
———————————1分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947521501.gif)
—————————————2分
在
中,由已知可得:
, 而![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947584665.gif)
,即
———————4分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947818435.gif)
——————————————————5分
(2)解:取
的中点
,连接![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947896535.gif)
由
为
的中点知
直线
与
所成的锐角就是异面直线
与
所成的角。 ——————6分
在
中,
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948114561.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948130183.gif)
是
斜边
上的中线
——————————————————————————8分
———————————————————————————10分
(3)解:设点
到平面
的距离为
。
———————————————————————— ———12分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948301815.gif)
在
中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231459483791150.gif)
而![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948395871.gif)
点
到平面的距离为
————————————————————————14分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947459241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947474502.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231459474902123.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947506344.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947521501.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947537344.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947552296.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947568547.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947584665.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947786495.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947802335.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947818435.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947833454.gif)
(2)解:取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947849234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947880327.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947896535.gif)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947412204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947974241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947989642.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948005128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948020241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948020365.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947396235.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947396242.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948098418.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948114622.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948114561.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948130183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948145372.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948176437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947849234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948208597.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948223651.gif)
(3)解:设点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947412204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947428270.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948270200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948286538.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948301815.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948317289.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948348626.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231459483791150.gif)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948395871.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948410804.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948005128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145947412204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145948457289.gif)
略
![](http://thumb2018.1010pic.com/images/loading.gif)
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