本题主要考查直线与平面平行的判定,以及平面与平面垂直的判定和三棱锥的体积的计算,体积的求解在最近两年高考中频繁出现,值得重视.
(1)欲证OD∥平面PAC,根据直线与平面平行的判定定理可知只需证OD与平面PAC内一直线平行,而OD∥PA,PA?平面PAC,OD?平面PAC,满足定理条件;
(2)欲证平面PAB⊥平面ABC,根据面面垂直的判定定理可知在平面PAB内一直线与平面ABC垂直,而根据题意可得PO⊥平面ABC;
(3)根据OP垂直平面ABC得到OP为三棱锥P-ABC的高,根据三棱锥的体积公式可求出三棱锥P-ABC的体积.
解:(Ⅰ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613808435.png)
分别为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613636492.png)
的中点,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613839200.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613683373.png)
∥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613870367.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613886406.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613699461.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613933462.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613699461.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613964395.png)
∥平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613699461.png)
. ………………5分
(Ⅱ)连结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614011377.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614042371.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614058686.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614073292.png)
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614104396.png)
中点,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613590485.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614136396.png)
⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614104396.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614167426.png)
.
同理,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614182364.png)
⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614104396.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614229412.png)
.
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614245538.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614276791.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614292679.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614448394.png)
⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614011377.png)
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614572239.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614182364.png)
⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614011377.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614182364.png)
⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614104396.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614697637.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614713391.png)
⊥平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613746473.png)
.
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614775235.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614791406.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613730441.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613839200.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613730441.png)
⊥平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613746473.png)
.…………………10分
(Ⅲ)由(Ⅱ)可知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614042371.png)
垂直平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613746473.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613839200.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231614042371.png)
为三棱锥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231613512534.png)
的高,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823231615446410.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408232316156491396.png)
. …………………………14分