题目
题型:不详难度:来源:
(Ⅰ)设P为AC的中点,Q在AB上且AB=3AQ,证明:PQ⊥OA;
(Ⅱ)求二面角O-AC-B的平面角的余弦值。
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015551-54907.jpg)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-54686.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-32992.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-20217.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015553-55557.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-32992.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015553-60506.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015553-31921.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015553-49586.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015553-89754.jpg)
在△ONB中,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-89197.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-62470.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-13093.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015555-22862.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015555-67946.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015555-40179.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015555-72705.jpg)
(Ⅱ)连结PN,PO.
由OC
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
又ON
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015555-40179.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
在等腰Rt△COA中,P为AC的中点,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
根据三垂线定理,知:AC
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015555-92310.gif)
在等腰Rt△COA中,OC="OA=1,"
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015556-96699.gif)
在Rt△AON中,ON=OA
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015556-91683.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015556-22962.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015556-80963.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015556-86796.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015554-44030.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015557-42923.gif)
解法二:
(Ⅰ)取O为坐标原点,以OA,OC所在的直线为x轴,z轴,建立空间直角坐标系O-xyz(如图所示)。
则A(1,0,0),C(0,0,1),B
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015557-31222.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015557-17371.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015557-51169.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015557-89687.gif)
又由已知,可得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015558-80577.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015558-30584.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015558-44595.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015558-35263.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015559-90106.gif)
(Ⅱ)记平面ABC的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015559-67588.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-69522.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015559-45480.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015559-45663.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015559-45480.gif)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015600-28094.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015600-67402.gif)
又平面OAC的法向量为e=(0,1,0)。
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015600-85250.gif)
二面角O-AC-B的平面角是锐角,记为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015600-17849.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015552-54686.gif)
核心考点
试题【如图,在四面体ABOC中,OC⊥OA。OC⊥OB,∠AOB=120°,且OA=OB=OC=1(Ⅰ)设P为AC的中点,Q在AB上且AB=3AQ,证明:PQ⊥OA;】;主要考察你对线线角等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-57872.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-31362.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-28048.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-43839.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-87299.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-43839.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-28778.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-60346.gif)
(1)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-42311.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-43839.gif)
(2)求平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015544-56492.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015543-43839.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015537-64283.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015537-93038.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015538-44932.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015538-25648.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015538-68617.gif)
A.30° | B.45° | C.60° | D.90° |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015531-82497.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015531-67490.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015531-31709.gif)
A.![]() | B.![]() | C.![]() | D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022015522-39347.gif)
①证明:PD// 面AGC;
②求AG和平面PBD所成的角的正切值。
![]() |
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