题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045214-91744.gif)
(1)求证:PB∥平面EFG;
(2)求异面直线EG与BD所成的角;
答案
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-61091.jpg)
∵E,F,G分别是线段PA、PD、CD的中点,
∴GH//AD//EF,
∴E,F,G,H四点共面。又H为AB中点,
∴EH//PB。又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-98751.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-88404.gif)
∴PB//面EFG。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-41893.gif)
(2)解:取BC的中点M,连结GM、AM、EM,则GM//BD,
∴∠EGM(或其补角)就是异面直线EG与BD所成的角。在Rt△MAE中,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-70314.gif)
同理
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045216-94752.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045216-48307.gif)
∴在△MGE中,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045216-69069.gif)
故异面直线EG与BD所成的角为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045216-40550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-41893.gif)
解法二:建立如图所示的空间直角坐标系A-xyz,
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045217-49564.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045217-89271.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045217-18796.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045217-94773.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045218-51535.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045218-84185.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045218-82740.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045218-65251.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045218-22872.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045219-18932.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045219-93398.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045219-67535.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045219-41806.gif)
解得
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-92656.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-78044.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-91215.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-51586.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-32174.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-91215.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045220-51586.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-88404.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-41893.gif)
(2)解:∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045221-24686.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045221-43731.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045221-52454.gif)
故异面直线EG与BD所成的角为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045216-40550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045215-41893.gif)
核心考点
试题【如图,平面PAD⊥平面ABCD,ABCD为正力形,∠PAD=900,且PA=AD=2,E、F、G分别是线段PA、PD、CD的中点。(1)求证:PB∥平面EFG;】;主要考察你对空间几何体的结构特征等知识点的理解。[详细]
举一反三
① 若a⊥α,b⊥α,则a∥b; ② 若 a∥α,b ∥α,则a∥b;
③ 若a⊥α,a⊥β,则α∥β; ④ 若α∥b,β∥b,则α∥β.
正确命题的个数是
A.1 | B.3 | C.2 | D.0 |
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045202-50954.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045202-69357.gif)
(Ⅰ)求异面直线GE与PC所成的角;
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045202-47542.jpg)
(Ⅲ)若F点是棱PC上一点,且DF⊥GC,求
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045203-84404.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-40788.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-94206.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-31753.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-74392.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-10397.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-46618.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-82474.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-80736.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045155-46618.gif)
(Ⅰ) 求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045156-54779.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045156-27318.gif)
(Ⅲ)求三棱锥B-PAC的体积.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045146-35378.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045147-97069.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045147-95621.gif)
A.3 | B.4 | C.5 | D.6 |
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045139-40145.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045140-86067.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045140-42237.jpg)
A.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045140-59594.gif)
B.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045140-90463.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045140-86067.gif)
C.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045141-57311.gif)
C.异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045141-44519.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045141-36855.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021045141-22337.gif)
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