题目内容
(本小题满分12分)
如图, 在直三棱柱ABC-A1B1C1中,AC=3,BC=4,
,AA1=4,点D是AB的中点.
(Ⅰ)求证:AC⊥BC1;
(Ⅱ)求二面角
的平面角的正切值.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514077075699.jpg)
如图, 在直三棱柱ABC-A1B1C1中,AC=3,BC=4,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407644274.gif)
(Ⅰ)求证:AC⊥BC1;
(Ⅱ)求二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407660357.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514077075699.jpg)
(Ⅰ)略
(Ⅱ)二面角
的正切值为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407785296.gif)
(Ⅱ)二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407769354.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407785296.gif)
(Ⅰ)证明:直三棱柱ABC-A1B1C1,底面三边长AC=3,BC=4,AB=5,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514078321244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514078471362.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407925541.gif)
∴ AC⊥BC, …………………2分
又 AC⊥
,且![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407956416.gif)
∴ AC⊥平面BCC1,又
平面BCC1 ……………………………………4分
∴ AC⊥BC1 ………………………………………………………………5分
(Ⅱ)解法一:取
中点
,过
作
于
,连接
…………6分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
是
中点,
∴
,又
平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408284324.gif)
∴
平面
,
又![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
平面
,
平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408284324.gif)
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408456331.gif)
∴
又![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
且![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408518402.gif)
∴
平面
,
平面
………8分
∴
又![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408066347.gif)
∴
是二面角
的平面角 ……………………………………10分
AC=3,BC=4,AA1=4,
∴在
中,
,
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408736319.gif)
∴
…………………………………………11分
∴二面角
的正切值为
…………………………………………12分
解法二:以
分别为
轴建立如图所示空间直角坐标系…………6分
AC=3,BC=4,AA1=4,
∴
,
,
,
,
∴
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409064427.gif)
平面
的法向量
, …………………8分
设平面
的法向量
,
则
,
的夹角(或其补角)的大小就是二面角
的大小 …………9分
则由
令
,则
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409251239.gif)
∴
………………10分
,则
……………11分
∵二面角
是锐二面角
∴二面角
的正切值为
………………………… 12分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514078321244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514078471362.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407925541.gif)
∴ AC⊥BC, …………………2分
又 AC⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407941255.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407956416.gif)
∴ AC⊥平面BCC1,又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407956282.gif)
∴ AC⊥BC1 ………………………………………………………………5分
(Ⅱ)解法一:取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407972241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407988204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408034210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408066347.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408066200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408097232.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408034210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408190235.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408206440.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408268255.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408284324.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408315264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408284324.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408346246.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408284324.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407956282.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408284324.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408456331.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408471348.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408066347.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408518402.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408534275.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408549281.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408346246.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408549281.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408596448.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408066347.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408643392.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407769354.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
∴在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408690405.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408456331.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408721439.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408736319.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408783983.gif)
∴二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407769354.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407785296.gif)
解法二:以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408830414.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408861362.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407910183.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408892350.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408955367.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408970355.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151408986407.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409002370.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409017593.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409064427.gif)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409080327.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409095387.gif)
设平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409126295.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409142534.gif)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409142220.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409158225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407660357.gif)
则由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231514091891331.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409204254.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409220364.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409251239.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409251493.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409314930.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151409345679.gif)
∵二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407769354.gif)
∴二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407769354.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823151407785296.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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