题目内容
如图,在正三棱柱ABC—A1B1C1中,底面边长为
,D为BC中点,M在BB1上,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231451397553015.jpg)
.
(1)求证:
;
(2)求四面体
的体积.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139724192.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231451397553015.jpg)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139771840.gif)
(1)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139786486.gif)
(2)求四面体
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139802338.gif)
(1)在正三棱柱ABC—A1B1C1中,D为BC的中点,
则AD
面BCC1B1,从而AD
MC …………(2分)
又
CM
AC1,则MC和平面ADC1内两相交直线AD,AC1均垂直,
MC
面ADC1, …………(4分)
于是MC
DC1. …………(6分)
(2)在矩形BB1C1C中,由CM
DC1知
DCC1~
MBC,设BB1=h,则BM=
h.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139880128.gif)
h:a=
:h,求得h=
.从而所求AA1=
. …………(8分)
连结![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140301865.gif)
…………(10分)
…………(12分)
则AD
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139818108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139818108.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139849183.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139818108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139880128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139818108.gif)
于是MC
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139818108.gif)
(2)在矩形BB1C1C中,由CM
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139818108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139942198.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139942198.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140098221.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145139880128.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140098221.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140145233.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140239260.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140239260.gif)
连结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140301865.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140317849.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823145140332948.gif)
略
![](http://thumb2018.1010pic.com/images/loading.gif)
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