题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022705-72591.png)
(Ⅰ)求证:平面O1DC⊥平面ABCD;
(Ⅱ)若∠A1AB=60°,求平面BAA1与平面CAA1的夹角的余弦值。
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-34549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-34221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-94911.png)
解析
试题分析:(Ⅰ)求证平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-55529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-68523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-23556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-68523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-22518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-24571.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-68523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-28741.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-77264.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-22518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-99138.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022708-99896.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-28741.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022708-19370.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-68523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-34549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-34221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022707-22518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022708-78515.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022708-34629.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022708-31083.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022709-64641.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022709-83033.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022709-54214.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022709-86023.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-63159.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-59209.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-34549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-34221.png)
试题解析:(Ⅰ)连结AC,BD, A1C1,则O为AC,BD的交点O1为A1C1,B1D1的交点。
由平行六面体的性质知:A1O1∥OC且A1O1=OC,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-88985.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-88985.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-88985.png)
又∵O1C
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-19805.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022710-88985.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-74336.jpg)
(Ⅱ)由题意可知Rt
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-86447.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-86447.png)
又∠A1AB=600,故
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-86447.png)
不妨设AB="a," 则在Rt
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-86447.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
如图分别以OB,OC,OA1为x轴,y轴,z轴建立空间直角坐标系,
则可得坐标为A(0,-
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-83274.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-69159.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022711-89034.png)
设平面ABA1的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-74905.png)
则由
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-83274.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-74905.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-55865.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-74905.png)
令x=1得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022712-74905.png)
又知BD⊥平面ACC1A1,故可得平面CAA1的一个法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022713-93709.png)
cosθ=|
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022713-99127.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-94911.png)
从而平面BAA1与平面CAA1的夹角的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022706-94911.png)
核心考点
试题【如图,已知平行六面体ABCD—A1B1C1D1的底面为正方形,O1、O分别为上、下底面的中心,且A1在底面ABCD上的射影是O。(Ⅰ)求证:平面O1DC⊥平面A】;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/20191106/20191106022659-41523.png)
(Ⅰ)求证:平面FGH⊥平面AEB;
(Ⅱ)在线段PC上是否存在一点M,使PB⊥平面EFM?若存在,求出线段PM的长;若不存在,请说明理由.
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022652-84865.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022652-17874.png)
A.若![]() ![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() |
①若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022645-84452.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022645-51753.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022646-42130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022646-13362.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022646-61643.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022646-69814.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022646-27573.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022647-97968.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022645-51753.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022647-55920.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022645-51753.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022647-59246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022647-97312.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022648-76152.png)
A.①④ | B.②③ | C.②④ | D.①③ |
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022635-34316.jpg)
(1)当E点是AB中点时,求证:直线ME‖平面ADD1 A1;
(2)若二面角AD1EC的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022635-68948.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022622-45555.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106022623-82269.jpg)
(Ⅰ)求证:AB⊥PD;
(Ⅱ)求证:GN//平面PCD.
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