题目内容
(本小题12分)
如图,已知
为平行四边形,
,
,
,点
在
上,
,
,
与
相交于
.现将四边形
沿
折起,使点
在平面
上的射影恰在直线
上.
(Ⅰ)求证:
平面
;
(Ⅱ)求折后直线DN与直线BF所成角的余弦值;
(Ⅲ)求三棱锥N—ABF的体积.![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315000132711278.gif)
如图,已知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000641303.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000641324.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000672442.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000688284.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000750204.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000844242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000859434.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000922335.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000953242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000969232.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000984211.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001000412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150000969232.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001047210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001078241.gif)
(Ⅰ)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001281266.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
(Ⅱ)求折后直线DN与直线BF所成角的余弦值;
(Ⅲ)求三棱锥N—ABF的体积.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315000132711278.gif)
(Ⅰ)
平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
(Ⅱ)
=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001421806.gif)
(Ⅲ)![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231500014371054.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001281266.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001374567.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001421806.gif)
(Ⅲ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231500014371054.gif)
解:(Ⅰ)
,得
面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001483281.gif)
则平面
平面
,
由
平面
平面
,
则
在平面
上的射影在直线
上,
又
在平面
上的射影在直线
上,
则
在平面
上的射影即为点
,
故
平面
. --------4分
(Ⅱ)法一.如图,建立空间直角坐标系,
∵在原图中AB=6,∠DAB=60°,
则BN=
,DN=2
,∴折后图中BD=3,BC=3
∴N(0,
,0),D(0,0,3),C(3,0,0)
=(-1,0,0)
∴
(-1,
,0)
(0,
,-3)
∴
=![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001421806.gif)
∴折后直线DN与直线BF所成角的余弦值为
-----9分
法二.在线段BC上取点M,使BM=BF,则MN∥BF
∴∠DNM或其补角为DN与BF所成角.
又MN=BF=2,DM=
.
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231500024351201.gif)
∴折后直线DN与直线BF所成角的余弦值为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002341267.gif)
(Ⅲ)∵AD∥EF, ∴A到平面BNF的距离等于D到平面BNF的距离,
∴![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231500014371054.gif)
即所求三棱锥的体积为
------14分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001452613.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001468253.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001483281.gif)
则平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001499313.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
由
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001561259.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001577332.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001047210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001655245.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001047210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001078241.gif)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001047210.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001842206.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001281266.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001062412.gif)
(Ⅱ)法一.如图,建立空间直角坐标系,
∵在原图中AB=6,∠DAB=60°,
则BN=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001967225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001967225.gif)
∴N(0,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001967225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002014531.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002123557.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001967225.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002217272.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001967225.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001374567.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150001421806.gif)
∴折后直线DN与直线BF所成角的余弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002341267.gif)
法二.在线段BC上取点M,使BM=BF,则MN∥BF
∴∠DNM或其补角为DN与BF所成角.
又MN=BF=2,DM=
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002419833.gif)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231500024351201.gif)
∴折后直线DN与直线BF所成角的余弦值为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002341267.gif)
(Ⅲ)∵AD∥EF, ∴A到平面BNF的距离等于D到平面BNF的距离,
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231500014371054.gif)
即所求三棱锥的体积为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150002482269.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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