题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-18406.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-84446.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-25583.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031826-79286.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-97869.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-66693.jpg)
上的点,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-25152.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-70162.gif)
(1) 求异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-87543.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-62126.gif)
(2) 证明
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-91096.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-49591.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-82091.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-39262.gif)
(3) 求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-88707.gif)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-46601.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-89016.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-55133.jpg)
点A为坐标原点,设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-98338.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-10002.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-68553.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-71367.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-66038.gif)
(1) 解:易得
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-43426.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-64245.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031829-73047.gif)
所以异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-87543.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031827-62126.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-46601.gif)
(2) 证明:已知
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031830-63357.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031830-12893.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031830-14441.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031830-43347.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031830-63194.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031830-43347.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031831-22847.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031831-27793.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031831-78777.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031831-92134.gif)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031832-25663.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-39262.gif)
(3)解:设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031832-45471.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031832-17286.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031832-85825.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031832-33015.gif)
不妨令X=1,可得
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031833-69412.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031833-76922.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031833-72606.gif)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031833-51504.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031833-16184.gif)
所以二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031834-61314.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-89016.gif)
方法二:(1)解:设AB=1,可得AD=2,AA1=4,CF=1.CE=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031834-88106.gif)
链接B1C,BC1,设B1C与BC1交于点M,易知A1D∥B1C,由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031834-95200.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031834-16448.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031835-86445.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031835-95202.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-46601.gif)
(2)证明:连接AC,设AC与DE交点N 因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031835-75001.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031835-57812.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031835-61241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031836-18428.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031836-22058.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031836-34288.gif)
连接BF,同理可证B1C⊥平面ABF,从而AF⊥B1C,所以AF⊥A1D因为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031836-71785.gif)
(3)解:连接A1N.FN,由(2)可知DE⊥平面ACF,又NF
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031837-24951.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031837-24951.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031837-38730.gif)
易知
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031837-15132.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031837-78146.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031837-51577.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031838-32926.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031838-56924.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031838-43712.gif)
连接A1C1,A1F 在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031838-48589.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031839-92395.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031839-50999.gif)
所以二面角A1-DE-F正弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031828-89016.gif)
核心考点
试题【如图,在长方体中,、分别是棱,上的点,,(1) 求异面直线与所成角的余弦值;(2) 证明平面(3) 求二面角的正弦值。】;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
A.(0,![]() | B.(1,![]() |
C.(![]() ![]() | D.(0,![]() |
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031813-74300.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031814-95511.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031814-44140.gif)
A.若![]() ![]() ![]() | B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() | D.若![]() ![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-80414.jpg)
本题满分15分)如图,在矩形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-27246.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-31113.gif)
在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-62590.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-70662.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-72186.gif)
将
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-28063.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031758-13818.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-92043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-33975.gif)
(Ⅰ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-46083.gif)
(Ⅱ)点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-16267.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-77718.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-71889.gif)
边形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-22102.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-60043.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031759-48778.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031800-30822.gif)
的长。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031750-99825.gif)
(Ⅰ)求证:FH∥平面EDB;
(Ⅱ)求证:AC⊥平面EDB;
(Ⅲ)求四面体B—DEF的体积;
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-83704.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-41305.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-48766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-41305.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031744-48766.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031745-90624.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021031745-65656.gif)
A.![]() | B.![]() |
C.![]() | D.![]() |
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