题目
题型:不详难度:来源:
如图,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082133-22413.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082134-67697.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082134-52553.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082134-83266.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082134-29518.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-32600.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-67258.jpg)
答案
∵AP=AB=2,BC=AD=
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-98022.gif)
∴A,B,C,D,P的坐标为A(0,0,0),B(2,0,0),C(2,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-98022.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-98022.gif)
又E,F分别是AD
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-10888.gif)
∴E(0,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-27853.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-27853.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-56241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082135-98022.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-16546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-27853.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-15157.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082137-42771.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-56241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-16546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-56241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-15157.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-56241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-16546.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-56241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082136-15157.gif)
∴PC⊥BF,PC⊥EF,BF ∩ EF=F,
∴PC⊥平面B
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082137-29028.gif)
(II)由(I)知平面BEF的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082137-25037.gif)
平面BAP 的法向量
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082137-73649.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082137-61695.gif)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082138-30184.gif)
∴ θ=45°, ∴ 平面BEF与平面BAP的夹角为45°.
解法二 (I)连接PE,EC在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082138-18742.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082138-24274.gif)
PA="AB=CD," AE=DE,
∴ PE=" CE," 即 △PEC 是等腰三角形,
又F是PC 的中点,∴EF⊥PC,
又
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082138-92462.gif)
∴ BF⊥PC.
又
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082139-41958.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082139-79180.gif)
(II)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082139-30377.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082139-80137.gif)
又ABCD是矩形,∴AB
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-28990.gif)
∴BC
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-28990.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-40422.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-28990.gif)
又由(Ⅰ)知PC
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-28990.gif)
∴ 直线PC与BC的夹角即为平面BEF与平面BAP的夹角,
在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-63750.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-16135.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082140-50968.gif)
所以平面BEF与平面BAP的夹角为45°.
解析
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082128-63014.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082128-99206.jpg)
(Ⅰ)证明:PC ⊥平面BEF;
(Ⅱ)求平面BEF与平面BAP夹角的大小.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082123-73125.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082124-64981.gif)
(I)求证BC
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082124-17829.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082124-75355.gif)
(III)设棱SA的中点为M,求异面直线DM与SB所成角的大小
如图,已知正三棱柱
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082117-36588.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082117-72943.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082118-49501.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082118-22216.jpg)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082118-57084.gif)
(Ⅱ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082118-85887.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082109-44078.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082109-11306.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082109-93604.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082109-58969.gif)
A
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-30500.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-78883.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-33790.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-33790.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-78883.gif)
C.若
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-16222.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-93082.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-93082.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082110-16222.gif)
在如图的多面体中,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-54702.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-39050.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-92826.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-99453.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-83364.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-76074.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082053-35933.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-97254.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-69667.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-45771.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-17139.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-41691.gif)
(Ⅰ) 求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-24588.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-89840.gif)
(Ⅱ) 求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-95238.gif)
(Ⅲ) 求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022082054-72387.gif)
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