解法一:(1)证明:连接
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804122380.png)
交
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804138402.png)
于
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804154302.png)
,连
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804169386.png)
. -------------------------------------2分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804154302.png)
为正方形
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804200534.png)
对角线的交点,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958042164595.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804154302.png)
为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804122380.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804138402.png)
的中点. ------------------------------------------------------3分
在D
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804372445.png)
中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195803998318.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804154302.png)
分别为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804013378.png)
、
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804450370.png)
的中点,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804169386.png)
//
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804512413.png)
. ----------------------------5分
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804169386.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804575205.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804512413.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804637276.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804029407.png)
//平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
. --------------------------7分
(2)解:设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804060342.png)
到平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
的距离为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805012321.png)
.
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805027561.png)
中,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805058582.png)
,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805074599.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805105721.png)
,
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958051211309.png)
, ----------------------------------------------------9分
于是
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958051521331.png)
. ----------------------------------------------------10分
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958051681776.png)
. --------------------------12分
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805199727.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805214903.png)
, --------------------------------------------13分
解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805230647.png)
,
故点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804060342.png)
到平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
的距离为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805292504.png)
. ----------------------------------------------------14分
解法二:(1)如图所示建立空间直角坐标系,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958053085074.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805339567.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805355603.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805386561.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805402590.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805417708.png)
,
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805433681.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805464835.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805480845.png)
----------------------------2分
设平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805495499.png)
的法向量
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805511652.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958055262096.png)
- --------------------------- ---------------------------3分
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958055421087.png)
,令
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805558354.png)
,则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805589444.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805604587.png)
-----------------------------4分
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958056201065.png)
,∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805698557.png)
, ------------------------6分
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804029407.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804637276.png)
平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804029407.png)
//平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
. ----------------------------7分
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805870843.png)
, ---------------------------------------------------------------9分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195805604587.png)
是平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
的一个法向量.
∴点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804060342.png)
到平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823195804044495.png)
的距离
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231958059631174.png)
.--------------------------------------14分