题目内容
(本题满分8分)
如图,在四棱锥
中,底面为直角梯形,
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315483436771.gif)
,
底面
,且
,
分别为
、
的中点。
(Ⅰ)求证:
;
(Ⅱ)求
与平面
所成角的正弦值。![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231548346795284.gif)
如图,在四棱锥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834320347.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834352333.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315483436771.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834461490.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834476246.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834492301.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834523595.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834539390.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834554236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834570234.gif)
(Ⅰ)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834586463.gif)
(Ⅱ)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834601242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231548346795284.gif)
(Ⅰ)略
(Ⅱ)
与平面
所成角的正弦值是
。
(Ⅱ)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834601242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834742309.gif)
(I)因为
是
的中点,
,所以
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231548348514157.jpg)
因为![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315483486685.gif)
平面
,所以
,从而
平面
.因为
平面
,所以
. (4分)
(II)取
的中点
,连结
、
,则
,
所以
与平面
所成的角和
与平面
所成的
角相等.
因为
平面
,所以
是
与平面
所成的角.
在
中,
,
故
与平面
所成角的正弦值是
。(8分)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834757211.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834570234.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834804308.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834835427.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231548348514157.jpg)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315483486685.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834882262.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834929266.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834944327.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834960254.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835007392.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834586463.gif)
(II)取
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835054236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835085213.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835100245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835132250.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835147335.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835100245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834601242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082315483525672.gif)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834960254.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835319412.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835100245.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835412466.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154835428839.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834601242.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834632437.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823154834742309.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
练习册系列答案
相关题目