题目内容
(本题14分)在如图所示的几何体中,
平面
,
平面![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356498479.png)
,且
,
是
的中点.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503566694155.png)
(I)求证:
;
(II)求
与平面
所成的角.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356467431.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356498479.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356560398.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356498479.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356591550.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356623799.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356638405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356654403.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503566694155.png)
(I)求证:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356685640.png)
(II)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356701478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
(I)证明见解析
(II)
.
(II)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356747369.png)
方法一:
(I)证明:因为
,
是
的中点,
所以
.
又
平面
,
所以
.
(II)解:过点
作
平面
,垂足是
,连结
交延长交
于点
,连结
,
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503571065536.png)
是直线
和平面
所成的角.
因为
平面
,
所以
,
又因为
平面
,
所以
,
则
平面
,因此
.
设
,
,
在直角梯形
中,
,
是
的中点,
所以
,
,
,
得
是直角三角形,其中
,
所以
.
在
中,
,
所以
,
故
与平面
所成的角是
.
方法二:
如图,以点
为坐标原点,以
,
分别为
轴和
轴,过点
作与平面
垂直的直线为
轴,建立直角坐标系
,设
,则
,
,
.
,
.
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503580425113.png)
(I)证明:因为
,
,
所以
,
故
.
(II)解:设向量
与平面
垂直,则
,
,
即
,
.
因为
,
,
所以
,
,
即
,
,
直线
与平面
所成的角
是
与
夹角的余角,
所以
,
因此直线
与平面
所成的角是
.
(I)证明:因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356763540.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356638405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356654403.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356810650.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356467431.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356498479.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356685640.png)
(II)解:过点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356638405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356997505.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357013308.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357028406.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357044410.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357059303.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357075482.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357091478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503571065536.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357122604.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356701478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356997505.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357215661.png)
又因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357215505.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357231562.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357262668.png)
则
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357278432.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357293555.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357387661.png)
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357403510.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357434779.png)
在直角梯形
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357449538.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357465649.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356638405.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356654403.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357512568.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357543715.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357590688.png)
得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357605602.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357621746.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503576371150.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357652673.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503576681021.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357683737.png)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356701478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356747369.png)
方法二:
如图,以点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357777319.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357793400.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357808399.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357824271.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357839313.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357777319.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356498479.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357871237.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357902535.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357403510.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357933632.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357933659.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357964681.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150357995676.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358027681.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503580425113.png)
(I)证明:因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358058795.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358089788.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358105759.png)
故
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358120642.png)
(II)解:设向量
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358136718.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358151545.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358167529.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358183656.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358198635.png)
因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358214807.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358245803.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358261477.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358261455.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358276624.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408231503583851434.png)
直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356701478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358432302.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358448295.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358463491.png)
所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150358479485.png)
因此直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356701478.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356716497.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823150356747369.png)
![](http://thumb2018.1010pic.com/images/loading.gif)
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