题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-36541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-57122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-88980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-61909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-90047.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-97410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-36361.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061420-24944.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-50467.png)
(Ⅰ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-39723.png)
(Ⅱ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-77664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-82242.png)
(Ⅲ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-17253.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-95708.png)
解析
试题分析:(1)根据题目中的线面的垂直性质定理得到线线垂直的证明。
(2)利用上一问的结论和线面垂直的判定定理得到证明。
(3)结合三垂线定理作出二面角的平面角,然后借助于三角形来求解大小。
(Ⅰ)证明:在四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-36541.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-57122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-88980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-69258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-88980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-42436.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-84304.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-98664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061423-66889.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061423-38375.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061423-66889.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061423-70168.png)
(Ⅱ)证明:由
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061423-87254.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061424-47645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061424-61548.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061424-81376.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-50467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061424-93050.png)
由(Ⅰ)知,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061424-62618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061424-21573.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-74996.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-32352.png)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-48114.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-32352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-32118.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-21332.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061426-10024.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-88980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061426-22166.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061426-18743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061426-36385.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061426-37111.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-77664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-82242.png)
(Ⅲ)解法一:过点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-59211.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-66108.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-14776.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-41453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-74996.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-32352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-70360.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061425-32352.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-41453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-22029.png)
因此
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-37162.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
由已知,得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-39618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-54466.png)
可得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-21466.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-38282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061429-21355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061429-49841.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061429-98450.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061429-98609.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061429-87487.png)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-95708.png)
解法二:由题设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-57122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061419-88980.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061430-59999.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061430-79986.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061430-80538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061430-22617.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061426-22166.png)
过点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061430-91559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061431-55534.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061431-44970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061431-39674.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061430-79986.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061431-44970.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061431-77653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061427-14776.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061431-56302.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061432-39698.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061432-89820.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
由已知,可得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-39618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061428-54466.png)
可得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061432-65045.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061432-66084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061432-26932.png)
于是,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061433-24256.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061433-32193.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061433-31708.png)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061421-14251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061422-95708.png)
(建立直角坐标系相应给分)
点评:解决该试题的关键是能合理的建立空间直角坐标系,表示出法向量以及直线的方向向量,借助于向量的知识来得到证明和求解,或者借助于线面的垂直的判定定理和性质定理得到结论。
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061408-67622.png)
A.![]() | B.![]() |
C.![]() ![]() ![]() | D.![]() ![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061410-88282.png)
A.若a![]() ![]() | B.若b![]() |
C.若a//α,α∩β=b则a//b | D.若a⊥α, b⊥α 则a//b |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061400-73889.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061400-98802.png)
①
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061401-88073.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061401-75884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061401-36428.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061401-39711.png)
其中正确的个数( )
A.1个 | B.2个 | C.3个 | D.4个 |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061350-86349.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061350-36243.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061350-19328.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061351-58152.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061351-12925.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061351-59118.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061351-17773.png)
(2)求
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061352-86715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022061352-22423.png)
A.垂直于同一个平面的两条直线互相平行 |
B.垂直于同一条直线的两个平面互相平行 |
C.如果平面![]() ![]() ![]() ![]() |
D.若平面![]() ![]() ![]() ![]() ![]() |
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