试题分析:(Ⅰ)求证:平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525545504.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525561477.png)
,证明面面垂直,先证线面垂直,即证一个平面过另一个平面的垂线,注意到F,H分别为线段PB,PC的中点,所以FH∥BC,只要CB⊥平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525561477.png)
,则FH⊥平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525561477.png)
,由已知EA⊥平面ABCD,则EA⊥CB,而四边形ABCD是正方形,CB⊥AB,从而可得CB⊥平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525561477.png)
,即可证出平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525545504.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525561477.png)
;(Ⅱ)这是一个探索性命题,一边假设存在,作为条件,进行推理即可,有已知条件,先判断EF⊥PB(因为若EF不垂直PB,则点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525639399.png)
就不存在),若PB⊥平面EFM,只需使PB⊥FM,注意到三角形
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525654448.png)
是一个直角三角形,这样△PFM∽△PCB,利用线段比例关系,可得PM=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525545488.png)
,从得结论.
试题解析:(Ⅰ)因为EA⊥平面ABCD,所以EA⊥CB.
又因为CB⊥AB,AB∩AE=A,所以CB⊥平面ABE. 3分
由已知F,H分别为线段PB,PC的中点,所以FH∥BC,则FH⊥平面ABE. 5分
而FH?平面FGH,所以平面FGH⊥平面ABE. 6分
(Ⅱ)在线段PC上存在一点M,使PB⊥平面EFM.证明如下:在直角三角形AEB中,因为AE=1,AB=2,所以BE=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525701322.png)
,
在直角梯形EADP中,因为AE=1,AD=PD=2,所以PE=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525701322.png)
,所以PE=BE.
又因为F为PB的中点,所以EF⊥PB...8分
要使PB⊥平面EFM,只需使PB⊥FM. ..9分
因为PD⊥平面ABCD,所以PD⊥CB,又因为CB⊥CD,PD∩CD=D,
所以CB⊥平面PCD,而PC?平面PCD,所以CB⊥PC.
若PB⊥FM,则△PFM∽△PCB,可得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525732844.png)
, 11分
由已知可求得PB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525748426.png)
,PF=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525763344.png)
,PC=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525779381.png)
,所以PM=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824032525545488.png)
..12分