题目内容
(本小题满分12分)
如图,正方形ADEF所在平面和等腰梯形所在平面AB
CD垂直,已知BC=2AD=4,
,![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058022324.gif)
(I)求证:
面ABF;
(II)求异面直线BE与AF所成的角;
(III)求该几何体的表面积。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231640580693429.jpg)
如图,正方形ADEF所在平面和等腰梯形所在平面AB
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316405797685.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058007493.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058022324.gif)
(I)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058038255.gif)
(II)求异面直线BE与AF所成的角;
(III)求该几何体的表面积。
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231640580693429.jpg)
(1)略
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058085239.gif)
(3)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058256838.gif)
⑴证明:因为面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058272409.gif)
面
,
交线
,
面
,
所以
面
. ……2分
故
,
又
,
.
所以
面
, ……4分
⑵解:注意到
,
所以
与
所成的角即为异面直线
与
所成的角, ……6分
连接
,由⑴知
.
在
中,
,
异面直线
与
所成的角为
. ……
8分
⑶解:由⑴知
面
,所以
,又
,
所以△
的面积
. ……9分
同理△
的面积
,等腰梯形
的上底长为
,下底长为4,两腰长均为
,则它的高为
,所以其面积
.……10分
等腰梯形
的上底长为
,下底长为4,两腰长均为
,
则它的高为
,所以其面积
.…… 11分
故该几何体的表面积
.…12分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058272409.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058288108.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303301.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303253.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058319236.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058334252.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058272409.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303253.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303301.gif)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058475315.gif)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058490324.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058506360.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058584258.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058600263.gif)
⑵解:注意到
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058615436.gif)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058631240.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058646241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058646241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058693229.gif)
连接
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058709244.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058724333.gif)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058756345.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231640587711310.gif)
异面直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058646241.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058693229.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058085239.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/2014082316405884972.gif)
⑶解:由⑴知
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303253.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303301.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059083318.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059099575.gif)
所以△
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058600263.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059130660.gif)
同理△
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059146275.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059161264.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059177409.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059192190.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059192255.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059224170.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059239729.gif)
等腰梯形
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164058303301.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059192190.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059192190.gif)
则它的高为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059317227.gif)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059348744.gif)
故该几何体的表面积
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823164059380838.gif)
![](http://thumb2018.1010pic.com/images/loading.gif)
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