题目
题型:不详难度:来源:
(1)求证: 平面ABD⊥平面ACD;
(2)求AD与BC所成的角;
(3)求二面角A—BD—C的大小.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021006-48403.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021006-82600.gif)
解析
∵平面ABC⊥平面BCD,∴AE⊥平面BCD,
∵BC⊥CD,由三垂线定理知AB⊥CD.
又∵AB⊥AC,∴AB⊥平面BCD,∵AB
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-61102.gif)
∴平面ABD⊥平面ACD。
(2)解: 在面BCD内,过D作DF∥BC,过E作EF⊥DF,交DF于F,由三垂线定理知AF⊥DF,∠ADF为AD与BC所成的角.
设AB=m,则BC=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-36002.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-55308.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-20797.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-40630.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021008-41519.gif)
即AD与BC所成的角为arctan
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021008-61623.gif)
(3)解:∵AE⊥面BCD,过E作EG⊥BD于G,连结AG,由三垂线定理知AG⊥BD,
∴∠AGE为二面角A—BD—C的平面角
∵∠EBG=30°,BE=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-55308.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021008-61211.gif)
又AE=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021007-55308.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021008-51641.gif)
即二面角A—BD—C的大小为arctan2.
另法(向量法): (略)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021009-49944.gif)
核心考点
试题【一副三角板拼成一个四边形ABCD,如图,然后将它沿BC折成直二面角.(1)求证: 平面ABD⊥平面ACD;(2)求AD与BC所成的角;(3)求二面角A—BD—C】;主要考察你对线线角等知识点的理解。[详细]
举一反三
(1)求证: AB1⊥C1D1;
(2)求证: AB1⊥面A1CD;
(3)若AB1=3,求直线AC与平面A1CD所成的角.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022021003-67395.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020951-44209.jpg)
如图,长方体
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020951-44549.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020951-55268.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020951-31460.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020952-69294.gif)
(1)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020952-10916.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020952-91969.gif)
(2)设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020952-18257.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020952-25933.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020953-49658.gif)
得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020953-13808.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020953-44649.gif)
在,求出
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020953-49658.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020942-91875.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020942-72886.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020942-93188.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020942-75421.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020943-67903.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020943-81071.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020943-34768.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020942-75421.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020943-47673.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020943-81071.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020937-14375.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020937-14375.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020937-14375.gif)
A.4条 | B.3条 | C.2条 | D.1条 |
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020929-61806.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020930-69909.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020930-45096.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020930-49955.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020930-63202.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020930-11911.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020931-81058.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020931-27293.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020931-86151.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022020931-83460.gif)
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