試題分析:(1)求證:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452845439.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452861481.png)
,證明線面垂直,先證線線垂直,即證線和平面內(nèi)兩條相交直線垂直,由已知可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453001556.png)
,只需證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453017596.png)
,或
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453033544.png)
,由已知平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452799496.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
,只需證明
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453095535.png)
,就得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453111422.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453017596.png)
,而由已知
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453158718.png)
,在直角梯形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
中,易求
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453204542.png)
,從而滿足
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453220696.png)
,即得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453095535.png)
,問題得證;(2)求二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的大小,可用傳統(tǒng)方法,也可用向量法,用傳統(tǒng)方法,關(guān)鍵是找二面角的平面角,可利用三垂線定理來找,但本題不存在利用三垂線定理的條件,因此利用垂面法,即作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453267571.png)
,與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453282385.png)
交于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453314302.png)
,過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453314302.png)
作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453345593.png)
,與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453360410.png)
交于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453392316.png)
,連結(jié)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453407397.png)
,由(1)知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453423552.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453454582.png)
,,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453470533.png)
是二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的平面角,求出
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453501506.png)
的三條邊,利用余弦定理,即可求出二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的大小,用向量法,首先建立空間坐標系,先找三條兩兩垂直的直線作為坐標軸,觀察幾何圖形可知,以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453532315.png)
為原點,分別以射線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453563552.png)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453579403.png)
軸的正半軸,建立空間直角坐標系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453594525.png)
,寫出個點坐標,設(shè)出設(shè)平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453610488.png)
的法向量為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453626768.png)
,平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453641447.png)
的法向量為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453657756.png)
,求出它們的一個法向量,利用法向量的夾角與二面角的關(guān)系,即可求出二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的大小.
(1)在直角梯形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
中,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453704541.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453719494.png)
得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453735665.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453158718.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453220696.png)
,即
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453095535.png)
,又平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452799496.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453111422.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453017596.png)
,又
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453001556.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452845439.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453922481.png)
;
(2)方法一:作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453267571.png)
,與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453282385.png)
交于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453314302.png)
,過點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453314302.png)
作
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453345593.png)
,與
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453360410.png)
交于點
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453392316.png)
,連結(jié)
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453407397.png)
,由(1)知,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453423552.png)
,則
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453454582.png)
,,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453470533.png)
是二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的平面角,在直角梯形
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
中,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454343691.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454359524.png)
,又平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452799496.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454421404.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454437473.png)
,從而,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454452521.png)
,由于
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453111422.png)
平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453064537.png)
,得:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454515540.png)
,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454546613.png)
中,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453719494.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454577545.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454593538.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534546083923.png)
在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454624634.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454640457.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454593538.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454671538.png)
,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454686596.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454702524.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454718485.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454593538.png)
,得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454764707.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454780710.png)
,從而
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454811606.png)
,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454827660.png)
中,利用余弦定理分別可得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534548421247.png)
,在
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453501506.png)
中,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534548891447.png)
,所以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053454905758.png)
,即二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的大小是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452923420.png)
.
方法二:以
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453532315.png)
為原點,分別以射線
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453563552.png)
為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453579403.png)
軸的正半軸,建立空間直角坐標系
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453594525.png)
如圖所示,由題意可知各點坐標如下:
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534550451957.png)
,設(shè)平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453610488.png)
的法向量為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453626768.png)
,平面
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453641447.png)
的法向量為
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053453657756.png)
,可算得
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053455139888.png)
,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534551541247.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534551861004.png)
得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534552011107.png)
,可取
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053455217827.png)
,由
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053455232949.png)
得,
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534552641157.png)
,可取
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053455279744.png)
,于是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534552951283.png)
,由題意可知,所求二面角是銳角,故二面角
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452877567.png)
的大小是
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/20140824053452923420.png)
.
![](http://thumb.zyjl.cn/pic2/upload/papers/20140824/201408240534553424503.png)
點評:本題主要考查空間點,線,面位置關(guān)系,二面角等基礎(chǔ)知識,空間向量的應(yīng)用 ,同時考查空間想象能力,與推理論證,運算求解能力.