题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053301-38539.png)
(Ⅰ)证明PQ⊥平面ABCD;
(Ⅱ)求异面直线AQ与PB所成的角;
(Ⅲ)求点P到平面QAD的距离.
答案
从而P、O、Q三点在一条直线上,所以PQ⊥平面ABCD.
(Ⅱ)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053301-99538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-98357.png)
解析
试题分析:(Ⅰ)连结AC、BD,设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-31146.png)
由P-ABCD与Q-ABCD都是正四棱锥,所以PO⊥平面ABCD,QO⊥平面ABCD.
从而P、O、Q三点在一条直线上,所以PQ⊥平面ABCD.
(Ⅱ)由题设知,ABCD是正方形,所以AC⊥BD.
由(Ⅰ),QO⊥平面ABCD. 故可分别以直线CA、DB、QP为x轴、y轴、z轴建立空间直角坐标系(如图),由题条件,相关各点的坐标分别是P(0,0,1),A(
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-10312.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-10312.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-92949.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053303-21544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053304-59445.png)
于是
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053304-19412.png)
从而异面直线AQ与PB所成的角是
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053301-99538.png)
(Ⅲ)由(Ⅱ),点D的坐标是(0,-
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-10312.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053304-19677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053304-17704.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053305-41215.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053305-91927.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053305-53490.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053305-27704.png)
取x=1,得
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053305-48354.png)
所以点P到平面QAD的距离
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053302-98357.png)
点评:典型题,立体几何题,是高考必考内容,往往涉及垂直关系、平行关系、角、距离的计算。在计算问题中,有“几何法”和“向量法”。利用几何法,要遵循“一作、二证、三计算”的步骤,利用向量则能简化证明过程。本题解法较多,特别是求角及距离时,运用了“向量法”,实现了问题的有效转化。对考生计算能力要求较高
核心考点
试题【已知两个正四棱锥P-ABCD与Q-ABCD的高分别为1和2,AB=4.(Ⅰ)证明PQ⊥平面ABCD; (Ⅱ)求异面直线AQ与PB所成的角;(Ⅲ)求点P到平面QA】;主要考察你对线线角等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053250-76696.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053251-83353.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053251-89503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053251-16777.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053251-50948.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053252-55014.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053252-97266.jpg)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053252-28471.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053252-28030.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053252-11578.png)
(Ⅱ)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053253-59256.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053253-40615.png)
如图,在四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053238-37315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053238-23479.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-42198.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-89122.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-87888.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-45373.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-28057.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-74544.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053239-62349.png)
(1) 设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053240-71973.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053240-10523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053240-20469.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053240-98725.png)
(2) 求四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053238-37315.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053231-23721.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053231-58591.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053231-20762.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053232-93008.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053232-85969.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053232-14137.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053232-23775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053232-59756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053233-53556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053233-41951.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053233-88456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053233-55307.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053234-63254.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053234-61145.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053234-92667.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053220-94317.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053220-51326.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053220-45979.png)
且
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053220-78307.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053220-80518.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053221-82081.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053221-68130.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053221-24602.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053221-76674.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053222-11419.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053222-97707.png)
(2)求截面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053222-97421.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053220-45979.png)
(3)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053222-35507.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053222-97421.png)
在如图的多面体中,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053204-48531.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053204-26110.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053204-75799.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053204-88001.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053205-20112.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053205-89565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053205-88886.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053205-65365.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053206-10743.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053206-26559.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053206-50797.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053206-27791.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053206-73064.png)
(Ⅱ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022053207-88684.png)
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