题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111417-89834.png)
(1)试确定点M的位置,使AC∥平面DMF,并说明理由;
(2)在(1)的条件下,求平面DMF与平面ABCD所成锐二面角的余弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111417-14786.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111418-35191.png)
解析
试题分析:(1)要使得AC∥平面DMF,需要使得AC平行平面DMF内的一条直线.为了找这条直线,需要作一个过AC而与平面DMF相交的平面.为此,连结CE,交DF于N,连结MN,这样只要AC∥MN即可.因为N为线段DF的中点,所以只需M是线段AE的中点即可.
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111418-51757.png)
(2)思路一、(综合法)首先作出它们的交线.过点D作平面DMF与平面ABCD的交线l,由于AC∥平面DMF,由线面平行的性质定理知AC∥l.为了求二面角,首先作出其平面角.作平面角第一步是过其中一个面内一点作另一个面的垂线,而要作垂线先作垂面.在本题中,由于平面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111418-41076.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111418-23973.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-25544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-37888.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-10498.png)
(1)当M是线段AE的中点时,AC∥平面DMF.
证明如下:
连结CE,交DF于N,连结MN,
由于M、N分别是AE、CE的中点,所以MN∥AC,
由于MN
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-95292.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111420-76435.png)
所以AC∥平面DMF. 4分
(2)方法一、过点D作平面DMF与平面ABCD的交线l,由于AC∥平面DMF,可知AC∥l,
过点M作MG⊥AD于G,
因为平面ABCD⊥平面CDEF,DE⊥CD,
所以DE⊥平面ABCD,则平面ADE⊥平面ABCD,
所以MG⊥平面ABCD,
过G作GH⊥l于H,连结MH,则直线l⊥平面MGH,所以l⊥MH,
故∠MHG是平面MDF与平面ABCD所成锐二面角的平面角. 8分
设
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111420-41592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111420-45103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111420-53567.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111421-82107.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111421-60102.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111421-34064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111418-35191.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111421-52071.png)
方法二、因为平面ABCD⊥平面CDEF,DE⊥CD,所以DE⊥平面ABCD,可知AD,CD,DE两两垂直,分别以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-25544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-37888.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111419-10498.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111422-61637.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111420-41592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111422-27858.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111422-62688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111422-14409.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111423-12537.png)
设平面MDF的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111423-91548.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111423-18206.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111423-88666.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111424-95534.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111424-13051.png)
取平面ABCD的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111424-29247.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111424-93038.png)
故平面MDF与平面ABCD所成锐二面角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111418-35191.png)
核心考点
试题【如图,四边形ABCD是梯形,四边形CDEF是矩形,且平面ABCD⊥平面CDEF,∠BAD=∠CDA=90°,,M是线段AE上的动点.(1)试确定点M的位置,使A】;主要考察你对空间向量的基本概念等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111410-56390.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111410-73028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111411-50530.png)
(1)试确定m,使直线AP与平面BDD1B1所成角为60º;
(2)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111411-79050.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111411-18970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111411-26038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111411-51473.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111353-37218.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111353-67752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111353-26534.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111353-67752.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-34295.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-84544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-74596.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-98187.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-72924.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-72751.png)
(1)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-45777.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-94074.png)
(2)在侧面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-19004.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-26313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111355-23673.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111355-16546.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111354-26313.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111355-71776.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111355-35260.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-55565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-63154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-87997.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-49475.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-70814.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-27210.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111334-50336.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-99364.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-82614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-61404.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-27499.png)
(1)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-25745.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-14906.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111335-85647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111317-43743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111317-79827.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111317-84965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111318-85956.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111318-79175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111318-87856.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111318-58243.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111318-85441.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111318-66640.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111319-18599.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111319-30555.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111319-72821.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111319-24472.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111319-21990.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111319-98878.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111320-42554.png)
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