题目
题型:不详难度:来源:
如图,已知四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-40529.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-21307.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-65306.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-26229.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-58966.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-92108.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-89284.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-34815.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-59095.gif)
(2)当平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-96290.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-63520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-48439.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-20772.gif)
答案
(1)证明:∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-21307.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-60024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-51686.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-41450.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-13412.gif)
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-65306.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-28432.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-30264.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-58966.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-33644.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-82788.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004700-43326.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-21307.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004700-35095.gif)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004700-57135.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-34815.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004700-50636.gif)
(2)解法1:由(1)知
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-34815.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-51686.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-34815.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004701-27005.gif)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-21307.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004701-72089.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004701-27005.gif)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004701-35837.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004701-90716.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-96290.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004701-63288.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004702-66366.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004702-12024.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004702-60083.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004702-31673.gif)
从而
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004703-28254.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004703-19049.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004703-23781.gif)
∴在等腰三角形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004703-17538.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-13412.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004704-43865.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004704-72877.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004704-13829.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004704-97386.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004704-21316.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004705-24601.gif)
∴中位线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004705-57411.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-21307.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004705-63002.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004705-56051.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004706-60823.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004706-81434.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004706-76316.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-48439.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004706-84468.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004706-64223.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004707-19642.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004707-19563.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004707-21358.gif)
∴二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-48439.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004707-60278.gif)
解法2:由(Ⅰ)知,以点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004707-36569.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-51686.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004659-13412.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004708-83448.gif)
所在的直线分别为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004708-86116.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004708-27286.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004708-46737.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004709-52482.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004709-29311.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004709-96821.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004709-74156.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004709-47565.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004710-15870.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004710-87938.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004710-55104.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004710-49736.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004710-60815.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004711-58933.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004711-47122.gif)
设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004711-72132.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004712-73535.gif)
则由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004712-94533.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004712-35134.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-96290.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004657-96794.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-63520.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004712-29587.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004712-55357.gif)
设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004713-16569.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004713-95903.gif)
则由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004713-39825.gif)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004712-35134.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004713-70792.gif)
设二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-48439.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004714-15546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004714-49389.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004714-82151.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004714-65825.gif)
∴二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004658-48439.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004714-75959.gif)
解析
核心考点
试题【(本小题满分12分)如图,已知四棱锥中,侧棱平面,底面是平行四边形,,,,分别是的中点.(1)求证:平面(2)当平面与底面所成二面角为时,求二面角的大小.】;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
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004645-81093.gif)
(I)求证:EF//平面ABC;
(II)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004645-23175.gif)
(III)求多面体ABDEC的体积。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004646-34720.jpg)
(1)求证:EF⊥平面BCD;
(2)求多面体ABCDE的体积;
(3)求平面ECD和平面ACB所成的锐二面角的余弦值。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004635-21657.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004620-33338.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004620-91428.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004620-12750.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004621-26279.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004621-48122.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004621-42182.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004621-10095.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004622-83433.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004622-99361.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004622-18358.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004622-71164.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004622-37999.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004623-21120.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004622-37999.gif)
(3)直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004623-37986.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004623-96982.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004623-12050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004623-53216.jpg)
B1C1上,CD=B1E=AC,ÐA
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004610-32140.gif)
求证:(1)BE∥平面AC1D;
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004610-93236.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004610-66215.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004559-15011.gif)
点O是△ABC的中心,将△DAO绕直线DO旋转一周,则在旋转过程中,直线DA与直线
BC所成角的余弦值的取值范围是 。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021004600-22347.gif)
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