题目
题型:不详难度:来源:
(1)求证AP∥平面EFG;
(2)求平面EFG与平面PDC所成角的大小;
(3)求点A到平面EFG的距离。
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041540-97835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041540-97835.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041540-10984.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041540-35070.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041541-83053.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041541-88161.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041541-29982.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041541-35129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041542-52431.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041542-47026.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041542-78075.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041542-21627.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041542-40785.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041543-43804.png)
设平面GEF的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041543-72525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041543-60338.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041543-84067.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041544-61799.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041544-40927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041544-21151.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041543-84067.png)
(Ⅱ)由(Ⅰ)知平面GEF的法向量 ,
因平面EFD与坐标平面PDC重合
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041544-62779.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041545-17337.png)
设平面间的夹角为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041545-48538.png)
故夹角的大小为45°。
(Ⅲ)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041545-62912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041545-34788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041546-19194.png)
解法二:(1)∵EF∥CD∥AB,EG∥PB,根据面面平行的判定定理
∴平面EFG∥平面PAB,又PA
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041546-10839.png)
(2)∵平面PDC⊥平面ABCD,AD⊥DC
∴AD⊥平面PCD,而BC∥AD,∴BC⊥面EFD
过C作CR⊥EF交EF延长线于R点连GR,根据三垂线定理知
∠GRC即为二面角的平面角,∵GC=CR,∴∠GRC=45°,
故平面间的夹角大小为45°。 (3)同上
解析
核心考点
试题【(本小题满分12)如图①在直角梯形ABCP中,BC∥AP,AB⊥BC,CD⊥AP,AD=DC=PD=2,E,F,G分别是线段PC、PD,BC的中点,现将ΔPDC】;主要考察你对向量与空间位置关系等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041528-59430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041529-88312.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041529-72506.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041529-13693.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041529-96487.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041530-33505.png)
(1)求证:面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041530-76175.png)
(2)求点C到平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041530-13207.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041530-64525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-31117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-55384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-91427.png)
(Ⅰ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-68686.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-74246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-90025.png)
(Ⅱ)若平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-16429.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-14743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041518-55384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041519-94072.png)
(Ⅲ)在棱
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041519-21510.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041519-64848.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041519-76229.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041519-23573.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041520-47316.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041520-95902.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041520-95170.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041504-10718.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041504-99268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041505-64318.png)
(1)求证:直线
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041505-44201.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041505-48593.png)
(2)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041505-98722.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041505-48593.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041506-29104.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041457-19511.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041457-95905.png)
A.![]() | B.![]() |
C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-54144.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-30850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-65906.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-88530.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-11305.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-92242.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-87429.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-15049.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106041451-65906.png)
A.①和② | B.③和④ | C.②和④ | D.①和③ |
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