题目内容
(本小题满分12分)
如图3,已知直二面角
,
,
,
,
,
,直线
和平面
所成的角为
.
(I)证明
;
(II)求二面角
的大小.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231502477331270.gif)
如图3,已知直二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247530357.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247546311.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247561263.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247592269.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247608318.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247624463.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247639234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247655201.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247670229.gif)
(I)证明
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247702358.gif)
(II)求二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247717325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231502477331270.gif)
(I)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247702358.gif)
(II)二面角
的大小为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247780411.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247702358.gif)
(II)二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247717325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247780411.gif)
解:(I)在平面
内过点
作
于点
,连结
.
因为
,
,所以
,
又因为
,所以
.
而
,所以
,
,从而
,又
,
所以
平面
.因为
平面
,故
.
(II)解法一:由(I)知,
,又
,
,
,所以
.
过点
作
于点
,连结
,由三垂线定理知,
.
故
是二面角
的平面角.
由(I)知,
,所以
是
和平面
所成的角,则
,
不妨设
,则
,
.
在
中,
,所以
,
于是在
中,
.
故二面角
的大小为
.
解法二:由(I)知,
,
,
,故可以
为原点,分别以直线
为
轴,
轴,
轴建立空间直角坐标系(如图).
因为
,所以
是
和平面
所成的角,则
.
不妨设
,则
,
.
在
中,
,
所以
.
则相关各点的坐标分别是
,
,
,
.
所以
,
.
设![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249184220.gif)
是平面
的一个法向量,由
得![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249308796.gif)
取
,得
.
易知
是平面
的一个法向量.
设二面角
的平面角为
,由图可知,
.
所以
.
故二面角
的大小为
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247795217.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247811205.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247826348.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247842209.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247842241.gif)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247858286.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247873387.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247889314.gif)
又因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247608318.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247920326.gif)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247936472.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247967476.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247967492.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247982360.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247826348.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248014276.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248092276.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248107261.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248092276.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248138354.gif)
(II)解法一:由(I)知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247982360.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247858286.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247873387.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248216309.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248279333.gif)
过点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247842209.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248310348.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248326206.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248341245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248357344.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248372306.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247717325.gif)
由(I)知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247889314.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248419301.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247639234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247655201.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248450483.gif)
不妨设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248466382.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248482319.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248497737.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248528366.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248575609.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248591413.gif)
于是在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248606376.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248622953.gif)
故二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247717325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248653415.gif)
解法二:由(I)知,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248669341.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248684344.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248716344.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247842209.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248731401.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248747187.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248762193.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248778129.gif)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248794299.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248419301.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247639234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247655201.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248450483.gif)
不妨设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248466382.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248482319.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248981280.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231502489961673.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248528366.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248575609.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150248591413.gif)
则相关各点的坐标分别是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249074360.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249090404.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249106386.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249121344.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249152485.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249168555.gif)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249184220.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249215432.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249230270.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249262728.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249308796.gif)
取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249355226.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249449509.gif)
易知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249480491.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247795217.gif)
设二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247717325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249589200.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249636477.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150249652950.gif)
故二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247717325.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150247780411.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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