题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021143-56242.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021144-79279.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021144-58466.gif)
点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021144-18445.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021144-80704.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021145-41052.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021145-45234.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021145-66831.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021145-51025.jpg)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-88748.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-71540.gif)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021144-18445.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-98719.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-82028.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-71540.gif)
(3)是否存在点
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021144-80704.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-62170.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-69313.jpg)
答案
(1)略
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-24636.gif)
(3)存在点E使得二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-62170.gif)
解析
(1)∵PA⊥底面ABC,∴PA⊥BC.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-96729.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021148-87306.gif)
∴BC⊥平面PAC.
(2)∵D为PB的中点,DE//BC,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021148-89379.gif)
又由(1)知,BC⊥平面PAC,
∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,
∵PA⊥底面ABC,∴PA⊥AB,又PA=AB,
∴△ABP为等腰直角三角形,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021148-97300.gif)
∴在Rt△ABC中,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021148-72923.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021148-29455.gif)
∴在Rt△ADE中,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-24636.gif)
(3)∵AE//BC,又由(1)知,BC⊥平面PAC,∴DE⊥平面PAC,
又∵AE
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021149-20421.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021149-20421.gif)
∴∠AEP为二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-62170.gif)
∵PA⊥底面ABC,∴PA⊥AC,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021149-97090.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021145-51025.jpg)
∴在棱PC上存在一点E,使得AE⊥PC,这时
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021149-86916.gif)
故存在点E使得二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021147-62170.gif)
解法2:如图,以A为原点建立空间直角坐标系
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021149-47225.gif)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021150-71342.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021150-40717.gif)
(1)∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021150-18314.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021145-51025.jpg)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021150-53894.gif)
又∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021148-87306.gif)
(2)∵D为PB的中点,DE//BC,∴E为PC的中点,
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021150-31096.gif)
∴又由(Ⅰ)知,BC⊥平面PAC,∴∴DE⊥平面PAC,垂足为点E.
∴∠DAE是AD与平面PAC所成的角,
∵
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021151-64556.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021151-33173.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-82028.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021146-71540.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021151-79563.gif)
核心考点
试题【(本小题满分14分)如图,在三棱锥中,底面,点,分别在棱上,且 (1)求证:平面;(2)当为的中点时,求与平面所成的角的正弦值;(3)是否存在点使得二面角为直】;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/20191021/20191021021006-93728.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021006-91252.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021006-64171.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021006-38526.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021006-60483.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021006-91049.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021007-23240.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021007-10095.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021007-66436.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021007-56236.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021007-50637.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021008-12655.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021008-49061.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021008-50013.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021008-86113.gif)
(2)求平面FGB与底面ABCD所成锐二面角的正切值。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021021008-10881.gif)
的平面β互相垂直,且
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020942-88259.gif)
BC=8,AB=6,若
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020942-94266.gif)
则点P在平面
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020942-91835.gif)
A.圆的一部分 | B.椭圆的一部分 |
C.双曲线的一部分 | D.抛物线的一部分 |
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020943-22271.jpg)
如图,三棱柱ABC—A1B1C1中,AA1⊥面ABC,BC⊥AC,BC=AC=2,AA1=3,D为AC的中点.
(Ⅰ)求证:AB1//面BDC1;
(Ⅱ)求二面角C1—BD—C的余弦值;
(Ⅲ)在侧棱AA1上是否存在点P,使得
CP⊥面BDC1?并证明你的结论.
如左图示,在四棱锥A-BHCD中,AH⊥面BHCD,此棱锥的三视图如下:
(1)求二面角B-AC-D的大小;
(2)在线段AC上是否存在一点E,使ED与面BCD成45°角?若存在,确定E的位置;若不存在,说明理由。
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020902-29325.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020903-27629.jpg)
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