题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111824-95225.png)
(1)证明:PF⊥FD;
(2)判断并说明PA上是否存在点G,使得EG∥平面PFD;
(3)若PB与平面ABCD所成的角为45°,求二面角A-PD-F的余弦值.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111824-76112.png)
解析
试题分析:解法一(向量法)
(I)建立如图所示的空间直角坐标系A-xyz,分别求出直线PF与FD的平行向量,然后根据两个向量的数量积为0,得到PF⊥FD;
(2)求出平面PFD的法向量(含参数t),及EG的方向向量,进而根据线面平行,则两个垂直数量积为0,构造方程求出t值,得到G点位置;
(3)由
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-33934.png)
解法二(几何法)
(I)连接AF,由勾股定理可得DF⊥AF,由PA⊥平面ABCD,由线面垂直性质定理可得DF⊥PA,再由线面垂直的判定定理得到DF⊥平面PAF,再由线面垂直的性质定理得到PF⊥FD;
(2)过点E作EH∥FD交AD于点H,则EH∥平面PFD,且有AH=
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-68701.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-68701.png)
(Ⅲ)由PA⊥平面ABCD,可得∠PBA是PB与平面ABCD所成的角,即∠PBA=45°,取AD的中点M,则FM⊥AD,FM⊥平面PAD,在平面PAD中,过M作MN⊥PD于N,连接FN,则PD⊥平面FMN,则∠MNF即为二面角A-PD-F的平面角,解三角形MNF可得答案..
试题解析:(1)证明:∵PA⊥平面ABCD,∠BAD=90°,AB=1,AD=2,建立如图所示的空间直角坐标系A-xyz,则
A(0,0,0),B(1,0,0),F(1,1,0),D(0,2,0).
不妨令P(0,0,t),∵
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-47947.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-14831.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111826-20764.png)
即PF⊥FD.
(2)解:设平面PFD的法向量为n=(x,y,z),
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111826-68507.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111826-85187.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111826-30937.png)
令z=1,解得:x=y=
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111827-33823.png)
∴n=
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111827-74920.png)
设G点坐标为(0,0,m),E
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111827-71503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111827-77516.png)
要使EG∥平面PFD,只需
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111828-99248.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111828-81222.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111828-69397.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-68701.png)
(3)解:∵AB⊥平面PAD,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-33934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111825-33934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111828-83641.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111829-79644.png)
故所求二面角A-PD-F的余弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111824-76112.png)
核心考点
试题【已知在四棱锥P-ABCD中,底面ABCD是矩形,且AD=2,AB=1,PA⊥平面ABCD,E、F分别是线段AB、BC的中点.(1)证明:PF⊥FD;(2)判断并】;主要考察你对空间向量的基本概念等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111816-43989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111817-18239.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111817-30945.jpg)
(1)求证:平面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111817-20557.png)
(2)若PA=AB=2,直线PB与平面EBD所成角的正弦值为
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111817-42818.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111803-62171.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111803-62171.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111803-71508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111803-44910.png)
(1)证明:DM
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111803-62171.png)
(2)求二面角A—DM—C的余弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111757-40518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111757-96274.png)
A.![]() |
B.3 |
C.6 |
D.9 |
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111749-94851.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111749-59839.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111750-34256.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111750-46965.png)
A.![]() |
B.![]() |
C.![]() |
D.![]() |
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-84036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-92521.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-52871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-52871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-22063.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-13574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-99363.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-91805.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111733-55908.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-57111.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-49653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-80230.png)
(1)设
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-49653.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-14452.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-35659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-36934.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111734-41668.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-53414.png)
(2)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-34564.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-65478.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-67568.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-24871.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-85253.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105111735-53547.png)
最新试题
- 1You’re standing in the dairy section at the grocery store, s
- 2解下列方程:(1)x2+6x=4;(2)2(x-3)2=3(x-3);(3)(x+8)(x+1)=-12.
- 3将20名城市义工(其中只有2名女性)平均分成两组,女性不在同一组的概率是( )A. B.
- 4【题文】选出下列句子中加点成语使用恰当的一项:A.“书山有路勤为径”,在知识爆炸的今天,我们更要努力攀登书山,而不能高山
- 5下列词语中加粗的字,读音全都正确的一组是[ ]A.饯别(jiàn)晦涩(huì)铁栅栏(zhà)浑身解数(xiè
- 6某文艺团体为希望工程组织了一场募捐义演,共售出1000张票,筹得票款6950元,已知成人票每张10元,学生票每张5元。(
- 7用如下装置可以完成一系列实验(图中夹持装置已略去)。 请回答下列问题:(1)若在装置A中选用浓硫酸和亚硫酸钠固体制取SO
- 8图示的西伯利亚地区是俄罗斯的一个重要开发区,这里针叶林广布…小题1:该地区的区域地理特征包括( )①广人稀 ②气
- 9读图。仔细观察下边的漫画,根据自己的理解,说说漫画的寓意。_______________________________
- 10已知方程(12)x=x13的解x∈(1n+1,1n),则正整数n=______.
热门考点
- 1如图,△ABC内接于⊙O,AD是△ABC的边BC上的高,AE是⊙O的直径,连接BE,△ABE与△ADC相似吗?请证明你的
- 2省力杠杆的动力臂______阻力臂.(选填“大于”、“小于”或“等于”)
- 3(6分)下列年代尺列举了某国历史上一些重点年代节点,根据提示回答问题A 1861年 B 1921年 C 1941年
- 4某同学用托盘天平称量镁粉25.2g(1g以下用游码),他把镁粉放在右盘,当天平平衡时,所称取的镁粉的实际质量是A.25.
- 5如图,已知直径MN⊥弦AB,垂足为C,下列结论:①AC=BC;②AN=BN;AM=BM;④AM=BM.其中正确的个数为(
- 6—What happened in Indonesia? —________ December, 2004, a str
- 7(理科做)已知函数f(x)=x3+ax+b定义在区间[-1,1]上,且f(0)=f(1).又P(x1•y1)、Q(x2•
- 8关于第一宇宙速度,以下说法中正确的是( )A.它是人造卫星绕地球飞行的最小速度B.它是地球表面附近发射人造卫星的最大发
- 9Once you _____ used to the weather here, you _____ to live h
- 10民族区域自治制度是我国的根本政治制度。