题目
题型:不详难度:来源:
是正三角形,且侧面PAD⊥底面ABCD,E为侧棱PD的中点.
(I)试判断直线PB与平面EAC的关系
(文科不必证明,理科必须证明);
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-65945.jpg)
(II)求证:AE⊥平面PCD;
(III)若AD=AB,试求二面角A-PC-D
的正切值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-17408.gif)
解析
(I)PB∥平面EAC.证明如下:
连结BD交AC于点O,连结EO,则O为BD的中点,
又∵E为PD的中点,∴EO∥PB,∴PB∥平面EAC.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-34208.jpg)
而侧面PAD
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-74924.gif)
∴CD⊥侧面PAD,∴CD⊥AE.
∵侧面PAD是正三角形,E为侧棱PD的中点,
∴AE⊥PD,∴AE⊥平面PCD;
(III)过E作EM⊥PC于M,连结AM,由(2)及三垂线定理知AM⊥PC.
∴∠AME为二面角A-PC-D的平面角. 10分
由正三角形PAD及矩形ABCD,且AD=AB,∴PD=AD=AB=DC,
∴在等腰直角三角形DPC中,设AB=a,则AE=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-40969.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041045-43782.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041045-35786.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041045-13369.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041045-66909.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041046-13332.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041046-16308.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-17408.gif)
即二面角A-PC-D的正切值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-17408.gif)
解法二:(I)同解法一
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041046-55800.jpg)
(II)设N为AD中点,Q为BC中点,则因为△PAD是正三角形,底面ABCD是矩形,所以,PN⊥AD,QN⊥AD,又因为侧面PAD⊥底面ABCD,所以,PN⊥面ABCD,QN⊥面PAD,以N为坐标原点,NA、NQ、NP所在直线分别为x,y,z轴如图建立空间直角坐标系.设AD=1,AB=a,则
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041046-31430.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041046-89318.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041047-99806.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041047-67002.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041047-62994.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041047-72064.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041048-31843.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041048-56437.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041048-38364.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041048-19700.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041048-91858.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041049-97246.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041049-73062.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041049-76070.gif)
∴AE⊥平面PCD;
(III)当a=1时,由(2)可知:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041048-31843.gif)
设平面PAC的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041049-20613.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041050-26592.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041050-61148.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041050-66529.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041050-15252.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041050-27094.gif)
向量
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-73937.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-39460.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-92018.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-14907.gif)
∴tanq=
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-17408.gif)
又由图可知,二面角A-PC-D的平面角为锐角,所以二面角A-PC-D的平面角就是向量
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-73937.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-39460.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041051-92018.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041044-17408.gif)
核心考点
试题【如图,四棱锥P-ABCD的底面是矩形,侧面PAD是正三角形,且侧面PAD⊥底面ABCD,E为侧棱PD的中点.(I)试判断直线PB与平面EAC的关系(文科不必证明】;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/20191021/20191021041028-86810.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041028-92667.gif)
(I)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041028-54031.gif)
(II)求直线PD与平面BCDE所成角的大小;
(III)求点D到平面PBC的距离.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041028-17520.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041013-40328.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041013-82962.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041014-16135.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041013-82962.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041014-72902.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041014-89692.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041014-48219.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041015-86908.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041015-88870.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041015-51470.jpg)
(I)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041014-48219.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041015-86908.gif)
(Ⅱ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041015-12352.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041002-73455.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041002-53296.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041003-95417.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041003-69439.gif)
(2)求三棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041003-87138.gif)
(3)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041003-36353.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021041003-86361.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-73017.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-18480.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-32703.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-29539.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-84983.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-26636.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-94254.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040946-67533.jpg)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040947-80152.gif)
(Ⅱ)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040947-58273.gif)
B沿棱柱侧面经过棱C C1到点A1的最短路线长为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040931-61238.gif)
点为D.
(1)求三棱柱ABC-A1B1C1的体积;
(2)在平面A1BD内是否存在过点D的直线与平面ABC平行?证明你的判断;
(3)证明:平面A1BD⊥平面A1ABB1.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021040931-49188.jpg)
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