(1)因为EA⊥平面ABC,AC?平面ABC,所以EA⊥AC,即ED⊥AC.
又因为AC⊥AB,AB∩ED=A,所以AC⊥平面EBD.
因为BD?平面EBD,所以AC⊥BD.
(2)因为点A,B,C在圆O的圆周上,且AB⊥AC,所以BC为圆O的直径.
设圆O的半径为r,圆柱高为h,根据正(主)视图,侧(左)视图的面积可得,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240508324221065.png)
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832438486.png)
所以BC=4,AB=AC=2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
.
以下给出求三棱锥E-BCD体积的两种方法:
方法一:由(1)知,AC⊥平面EBD,
所以V
E-BCD=V
C-EBD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832469276.png)
S
△EBD×CA,
因为EA⊥平面ABC,AB?平面ABC,
所以EA⊥AB,即ED⊥AB.
其中ED=EA+DA=2+2=4,
因为AB⊥AC,AB=AC=2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
,
所以S
△EBD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832516274.png)
ED×AB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832516274.png)
×4×2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
=4
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
,
所以V
E-BCD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832469276.png)
×4
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
×2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832407319.png)
.
方法二:因为EA⊥平面ABC,
所以V
E-BCD=V
E-ABC+V
D-ABC=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832469276.png)
S
△ABC×EA+
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832469276.png)
S
△ABC×DA=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832469276.png)
S
△ABC×ED.
其中ED=EA+DA=2+2=4,
因为AB⊥AC,AB=AC=2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
,
所以S
△ABC=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832516274.png)
×AC×AB=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832516274.png)
×2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
×2
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832454365.png)
=4,
所以V
E-BCD=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832469276.png)
×4×4=
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824050832407319.png)
.