题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025718-98366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025718-30258.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025718-64854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025719-33491.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025719-18242.png)
(1)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025719-10893.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025719-24732.png)
(2)当
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-57479.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-87564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025718-64854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-44651.png)
(3)在(2)的条件下若F是PD的靠近P的一个三等分点,求二面角A-EF-D的余弦值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-12990.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-49485.png)
解析
试题分析:(1)证面面垂直,先证明线面垂直.那么证哪条线垂直哪个面?因为ABCD是正方形,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-49897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-50540.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-35715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-17538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-49028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-28885.png)
(2)设AC交BD=O.由(1)可得
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-96886.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-39286.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025723-15793.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025723-72597.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-87564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-39286.png)
(3)在本题中作二面角的平面角较麻烦,故考虑建立空间直角坐标系,然后用空间向量求解.
试题解析:(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025723-38041.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-49897.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-50540.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-35715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025723-86421.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-35715.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-17538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025724-41412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-49028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025722-28885.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025723-86421.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025724-99918.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025719-10893.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025719-24732.png)
(2) 设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025724-27956.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025724-88075.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025723-72597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025725-56968.png)
在直角三角形ADB中,DB=PD=2,则PB=
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025725-88959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025725-57843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025725-88959.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025725-55335.png)
即E为PB的中点.
(3) 连接OE,因为E为PB的中点,所以
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025726-19217.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025721-35715.png)
则A(1,0,0), E(0,0,1) ,F(0,-1,
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025726-22697.png)
平面EFD的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025726-78827.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025726-27460.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025726-23591.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025727-93559.png)
令y=1,则
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025727-74939.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025727-11155.png)
所以二面角A-EF-D的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025720-49485.png)
核心考点
试题【如图,四棱锥的底面是正方形,,点在棱上.(1)求证:平面平面;(2)当,且时,确定点的位置,即求出的值.(3)在(2)的条件下若F是PD的靠近P的一个三等分点,】;主要考察你对向量与空间位置关系等知识点的理解。[详细]
举一反三
①
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025709-36465.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025709-48525.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025710-68146.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025710-21648.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025710-78381.jpg)
A.1 | B.2 | C.3 | D.4 |
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025655-97776.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025655-99652.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025655-99652.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025655-97776.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025655-85063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025655-28052.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-88240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-10186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-88589.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-88125.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-81868.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-40112.png)
(Ⅱ)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-95912.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-26043.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-12032.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-28727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025656-39957.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025657-76126.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025632-94608.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025632-21936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025632-71457.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025632-21936.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025632-42569.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025632-69539.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-27788.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-23503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-14961.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-49888.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-23866.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-55084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-71202.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-22562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-62035.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025633-45190.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025634-67078.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025618-69241.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025618-97194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025618-72018.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025619-12867.png)
A.若![]() ![]() ![]() | B.若![]() ![]() ![]() |
C.若![]() ![]() ![]() | D.若![]() ![]() ![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025601-87164.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025601-91467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025602-39362.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025602-41536.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025602-54624.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025601-87164.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025602-88590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025603-82384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025603-18488.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025603-88280.png)
(1)求异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025603-61252.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025604-45016.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191106/20191106025604-88301.png)
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