题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113826-22197.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113826-58555.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113826-27650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-81438.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-54741.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-61709.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-61709.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113826-22197.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-73745.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-95558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-82689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-33170.png)
(1)当
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-63434.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-30275.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-88715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-86958.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-23118.png)
解析
试题分析:(1)因为当
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-63434.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-94432.png)
(2)由题意所给的体积关系可确定点G的位置,求二面角关键是转化为两平面的法向量的夹角,由于平面BCG的法向量易得,关键是求出平面DGB的法向量.通过待定系数法即可求得,还需判断二面角与法向量夹角的大小关系.解法二用到的推理论证的数学思想很重要.
试题解析:(1)证明:∵点
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-42353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-89078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-10297.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-38960.png)
又∠ABC=90°∴AE⊥EF,∵平面AEFD⊥平面EBCF,
∴AE⊥平面EBCF,AE⊥EF,AE⊥BE, 又BE⊥EF,
如图建立空间坐标系E﹣xyz.
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-82956.png)
翻折前,连结AC交EF于点G,此时点G使得AG+GC最小.
EG=
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-61440.png)
则A(0,0,2),B(2,0,0),C(2,4,0), D(0,2,2),E(0,0,0),G(0,2,0),
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-22378.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-91576.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-42137.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-68188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-77334.png)
(2)解法一:设EG=k,
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113829-13706.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-69211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-68936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-70442.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-81408.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-27491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-88093.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-62847.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-72233.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-49803.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113831-36593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-68936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113831-67791.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-88093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113831-32493.png)
设平面DBG的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113831-59763.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113831-31226.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113831-52769.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113832-48479.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113832-18185.png)
取x=1,则y=2,z=-1,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113832-65888.png)
面BCG的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113832-42912.png)
则cos<
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113833-50538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113833-54966.png)
所以此二面角平面角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-23118.png)
(2)解法二:由解法一得EG=1,过点D作DH
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-95558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113833-43210.png)
∵平面AEFD⊥平面EBCF,
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-68936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-95558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113830-68936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113827-95558.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113833-79499.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113834-37922.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113834-94650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113834-91878.png)
又DH=2,在
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113834-94650.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113834-29499.png)
所以此二面角平面角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113828-23118.png)
核心考点
举一反三
①(a·b)2=a2·b2;②|a+b|>|a-b|;③|a+b|2=(a+b)2;④若a∥b,则a·b=|a|·|b|.其中真命题的个数是( )
A.1 | B.2 | C.3 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113802-90848.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113802-27896.png)
(1)若点E在SD上,且
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113802-60996.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113803-67254.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113803-63677.png)
(2)若三棱锥S-ABC的体积
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113803-35352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113752-87428.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113753-63527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113753-56570.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113753-45873.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113753-78907.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113744-68317.png)
(1)求证BC⊥平面AFG;
(2)求二面角B-AE-D的余弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113734-36002.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113734-10184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113734-57030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113734-77983.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113734-66654.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113735-17881.png)
(1)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113735-10540.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113735-84274.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113735-96245.png)
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