题目
题型:不详难度:来源:
如图,在三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033357-39988.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033357-28296.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033357-17177.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033357-50400.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033358-16668.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033358-77441.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033358-96740.png)
(Ⅱ)若异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033358-24696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033359-52707.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033359-52754.png)
答案
(Ⅱ)二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033359-80189.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033359-28670.png)
解析
(1)利用先棉农垂直的性质定理得到线线垂直的证明即可。
(2)建立空间直角坐标系,然后表示出平面的法向量和法向量的夹角,即为二面角的平面角的求解。
解:(Ⅰ)证明:∵ 面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-31987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033357-17177.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033358-16668.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-31987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-58232.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033357-17177.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-62484.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-81311.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-31987.png)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-20398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033400-31987.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033401-90560.png)
(Ⅱ)取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033401-98754.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033401-73559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033401-49801.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033402-99336.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033402-90693.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033401-73559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033402-86878.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033403-50134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033403-64772.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033403-12007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033403-46084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033404-39346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033404-79857.png)
根据
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033404-74267.png)
可得平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033404-69335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033404-98153.png)
而平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033405-81445.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033405-19735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033405-14822.png)
因此,二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033359-80189.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033359-28670.png)
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033352-92019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033352-73645.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-95556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-80430.png)
①若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-86240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-61722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-34815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-34815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-86240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033353-61722.png)
③若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033354-92492.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033354-81741.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033354-43986.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033354-81741.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033354-43986.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033354-92492.png)
其中为真命题的序号是_______
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033339-24279.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033339-61467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033339-38038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033340-30038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033340-68185.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033340-13188.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033340-59465.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033341-95449.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033341-62224.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033341-94540.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033341-30695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033341-26020.png)
⑴证明
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033342-10649.png)
⑵求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033342-95631.png)
已知斜三棱柱ABC—A1B1C1的底面是正三角形,侧面ABB1A1是边长为2的菱形,且
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033327-39818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033327-29531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033328-91753.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033328-32509.png)
(2)求点M到平面AA1C1C的距离.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033316-86173.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033316-28550.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033316-34007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033316-38562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033317-36099.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033317-53426.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033317-33688.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033317-47592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033318-72504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033318-93099.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033318-52541.png)
(Ⅰ)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033318-13272.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033319-47942.png)
(Ⅱ)求
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033319-13306.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033319-45494.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033300-72904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033300-63435.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033301-17204.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033301-89125.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033301-23667.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033301-72720.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033302-26015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033302-41401.jpg)
(I)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033302-90805.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033301-17204.png)
(II)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033302-90790.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033303-78000.png)
(III)在
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033303-36695.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033303-59672.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033303-20120.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033303-78000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106033303-59672.png)
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