题目
题型:不详难度:来源:
(Ⅰ)若在边BC上存在一点Q,使PQ⊥QD,求a的取值范围;
(Ⅱ)当边BC上存在唯一点Q,使PQ⊥QD时,求二面角A-PD-Q的余弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-77081.gif)
答案
(1)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-83065.gif)
(2)二面角A-PD-Q的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-86334.gif)
解析
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-24071.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020116-16420.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020116-98697.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020116-47455.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020116-12466.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020117-73655.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020117-15464.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020117-69631.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020117-99477.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020117-24926.gif)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-54524.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-20937.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-80960.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-80962.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-37702.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-48315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-83065.gif)
(Ⅱ)由(Ⅰ)知,当
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-65472.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-96998.gif)
使PQ⊥QD.
过Q作QM∥CD交AD于M,则QM⊥AD.
∵PA⊥平面ABCD,∴PA⊥QM.∴QM⊥平面PAD.
过M作MN⊥PD于N,连结NQ,则QN⊥PD.
∴∠MNQ是二面角A-PD-Q的平面角. ……10分
在等腰直角三角形
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-77593.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020120-96701.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020120-63118.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020120-96932.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020120-79448.gif)
故二面角A-PD-Q的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-86334.gif)
解法2:(Ⅰ)以
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020120-86622.gif)
B(0,2,0),C(a,2,0),D(a,0,0),
P(0,0,4), ……2分
设Q(t,2,0)(
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020121-23659.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020121-16201.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020121-77664.gif)
∵PQ⊥QD,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020121-34396.gif)
即
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-80962.gif)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020118-37702.gif)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-48315.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-83065.gif)
(Ⅱ)由(Ⅰ)知,当
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-65472.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-96998.gif)
此时Q(2,2,0),D(4,0,0).
设
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020122-83282.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020122-60729.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020122-20730.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020122-97354.gif)
取
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020123-67149.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020123-19428.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020122-60729.gif)
而
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020123-44809.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020119-77593.gif)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020123-69616.gif)
∴二面角A-PD-Q的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020115-86334.gif)
核心考点
试题【 如图,在矩形ABCD中,AB=2,BC=a,又PA⊥平面ABCD,PA=4. (Ⅰ)若在边BC上存在一点Q,使PQ⊥QD,求a的取值范围;(Ⅱ)当边B】;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/20191021/20191021020053-65651.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-60343.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-46099.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-70817.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-38138.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-59570.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020053-65651.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-60343.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020055-24452.gif)
(1)试判断直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-46099.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020055-91746.gif)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020055-88396.gif)
(3)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020054-59570.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020055-49369.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020056-46550.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020056-98752.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020056-58471.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020039-15558.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020040-20951.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020040-65743.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020040-18817.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020040-66404.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020040-78172.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020041-27168.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020041-86049.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020041-39688.gif)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020041-37442.gif)
(Ⅱ)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020042-31729.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020042-55197.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020042-64582.gif)
如图,四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-13341.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-54241.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-75615.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-83270.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-48567.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-70935.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-55298.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020031-72121.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020032-55253.gif)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020032-34124.gif)
(2)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020032-80777.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020032-98217.gif)
(3)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020032-35939.gif)
已知正方形ABCD的边长为1,
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020017-27122.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020017-31848.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020017-31918.gif)
(I)若点M是棱AB的中点,求证:OM∥平面ACD;
(II)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020017-58646.gif)
(III)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020018-95524.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020018-14381.jpg)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020018-86345.gif)
![](http://img.shitiku.com.cn/uploads/allimg/20191021/20191021020018-87276.jpg)
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