题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
(1)求直线A′C与DE所成的角;
(2)求直线AD与平面B′EDF所成的角;
(3)求面B′EDF与面ABCD所成的角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-30060.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-89564.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-80197.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-49385.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-51582.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-58442.gif)
由余弦定理得cosA′CP=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-54319.gif)
故A′C与DE所成角为arccos
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-54319.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
(2)解
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-89564.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-12528.gif)
在Rt△B′AD中,AD=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-10952.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-10952.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-10952.gif)
则cosADB′=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-72481.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-72481.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-53818.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-89564.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
在Rt△DOE中,OE=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-30606.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-86363.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-51582.gif)
则由面积关系得OM=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-35448.gif)
在Rt△OHM中,sinOMH=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-93439.gif)
故面B′EDF与面ABCD所成的角为arcsin
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-45660.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
方法二(向量法)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-89564.gif)
(1) 如图建立坐标系,则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-82582.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044024-37300.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044025-50937.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044025-25792.gif)
故A′C与DE所成角为arccos
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-54319.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
(2)∵∠ADE=∠ADF,∴AD在平面B′EDF内的射影在∠EDF的平分线上
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
故直线AD与平面B′EDF所成的角为∠ADB′,如图建立坐标系,则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044025-42735.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044025-43870.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044026-46805.gif)
故AD与平面B′EDF所成的角是arccos
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044023-72481.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
(3)由(1)知
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044026-66086.gif)
所以面ABCD的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-89564.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044026-61879.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044026-83566.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-89564.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044026-65673.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044027-83592.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044027-72177.gif)
取z=1,得
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044027-78817.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044027-15653.gif)
故面B′EDF与面ABCD所成的角为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044028-92237.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044022-32061.gif)
核心考点
试题【在棱长为a的正方体ABCD—A′B′C′D′中,E、F分别是BC、A′D′的中点 (1)求直线A′C与DE所成的角;(2)求直线AD与平面B′EDF所成的角;】;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
(II)求证:AC 1//平面CDB1;
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044007-19473.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044002-52304.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044002-52304.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021044002-11631.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043948-67223.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043948-12869.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043948-42317.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043948-71199.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-18529.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-42975.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-40315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-35475.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-94890.gif)
(2)证明
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-99478.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-37618.gif)
(3)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043949-21460.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043933-84291.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-69989.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-21432.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-95658.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-18428.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-64514.gif)
(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-34439.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043921-73566.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043922-67468.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021043922-17391.gif)
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