题目内容
如图1,在直角梯形
中,
,
,
,
,
分别是
的中点,现将
沿
折起,使平面
平面
(如图2),且所得到的四棱锥
的正视图、侧视图、俯视图的面积总和为8.
⑴求点
到平面
的距离;
⑵求二面角
的大小的夹角的余弦值;
⑶在线段
上确定一点
,使
平面
,并给出证明过程.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630195765091.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019201518.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019217585.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019232527.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019248560.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019264620.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019279530.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019279613.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019310531.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019326405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019342481.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019357526.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019373603.png)
⑴求点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019404313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019420477.png)
⑵求二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019435580.png)
⑶在线段
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019466365.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019498333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019529413.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019544490.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630195765091.png)
(1)二面角
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019435580.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019622356.png)
(2)点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019498333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019466365.png)
解:(1)由几何体的正视图、侧视图、俯视图的面积总和为8可得
,取
中点
,联结
,
分别是
的中点,
,∴
四点共面.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316301995030758.png)
作
于
,易得:
平面
且
.
又
平面
,故点
到平面
的距离
即为所求.
(2)
就是二面角
的平面角
在
中,
, ![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020402724.png)
,即二面角
的大小为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019622356.png)
解法二:建立如图所示空间直角坐标系,设平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020668489.png)
的一个法向量为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020683631.png)
则![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630207141243.png)
取
,又平面
的法向量为
(1,0,0)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630210891456.png)
(3)设
则![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630212921144.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630213231944.png)
又
平面
点
是线段
的中点.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019685688.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019716385.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019732303.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019747525.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019763540.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019279613.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019810772.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019841596.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316301995030758.png)
作
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019966586.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019498333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020090439.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020122546.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020137652.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020153467.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020122546.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020200315.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020122546.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020137652.png)
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020262548.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019435580.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020293618.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020324722.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020402724.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020418676.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019435580.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019622356.png)
解法二:建立如图所示空间直角坐标系,设平面
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020668489.png)
的一个法向量为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163020683631.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630207141243.png)
取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163021011556.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163021026472.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163021073350.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630210891456.png)
(3)设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163021276929.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630212921144.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231630213231944.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163021338693.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163021354858.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019498333.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823163019466365.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目