题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032353-60812.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032353-61325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032353-27890.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032354-98574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032354-22397.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032354-90377.jpg)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032354-68546.png)
(2)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032354-97479.png)
(3)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032354-70278.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-84915.png)
解析
(1)yw由于
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-63566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-20127.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-45068.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-84728.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-11084.png)
(2) 取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-40201.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-30001.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-72658.png)
由条件知
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-75827.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-68083.png)
∴四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-88677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-88926.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-37446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-21725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-10660.png)
∴四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-62457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-16924.png)
然后得到结论。
(2)建立空间直角坐标系,然求解平面的法向量的坐标,结合向量的数量积的性质得到夹角的值。
证明:(Ⅰ)由于
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-63566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-20127.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-45068.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-84728.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-11084.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032359-34545.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032359-19965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032359-87338.png)
(Ⅱ)取
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-40201.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-30001.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-72658.png)
由条件知
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032356-75827.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-68083.png)
∴四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-88677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-88926.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032357-37446.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-21725.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-10660.png)
∴四边形
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-62457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032358-16924.png)
∴平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032359-36335.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032400-70778.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032400-48663.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032400-70778.png)
(Ⅲ)由(Ⅰ)知
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032400-54225.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032400-69353.jpg)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032401-61060.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032401-44603.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032401-37899.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032401-58539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032402-14881.png)
设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032400-70778.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032402-22815.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032402-22106.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032402-25009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032402-57883.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032403-96835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032403-26093.png)
而平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032403-88085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032403-26836.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032404-88752.png)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032404-63912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032404-63912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032355-84915.png)
核心考点
举一反三
A.平面![]() ![]() ![]() ![]() |
B.若直线![]() ![]() ![]() |
C.若平面![]() ![]() ![]() ![]() ![]() ![]() |
D.若直线![]() ![]() ![]() |
如图(1)在直角梯形ABCP中,BC∥AP,AB⊥BC,CD⊥AP,AD=DC=PD=2,E、F、G分别是PC、PD、BC的中点,现将△PDC沿CD折起,使平面PDC⊥平面ABCD(如图2)
(1)求二面角G-EF-D的大小;
(2)在线段PB上确定一点Q,使PC⊥平面ADQ,并给出证明过程.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032334-59535.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032327-97646.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032327-46071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-66009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-97937.png)
①设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032327-97646.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032327-46071.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-66009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-97937.png)
②设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-61965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-90989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-49154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-66009.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-97937.png)
③设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032329-34374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032329-30024.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032329-47454.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-90989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032329-61848.png)
④设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032330-58490.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032330-19226.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032328-90989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032329-61848.png)
其中正确命题的个数是( )
A.![]() | B.![]() | C.![]() | D.![]() |
如图,在三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032312-62915.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032312-85251.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032312-82015.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032312-26402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032313-32109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032313-20758.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032313-89354.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032313-39268.png)
(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032314-33816.png)
(2)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032314-48987.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032314-85788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032314-12891.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032315-40707.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032315-35354.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032306-31870.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032306-87891.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032306-25205.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032306-73917.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032306-66807.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032306-35700.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106032307-65555.png)
A.0个 | B.1个 | C.2个 | D.3个 |
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