题目
题型:不详难度:来源:
如图,在四棱锥S - ABCD中,底面ABCD是直角梯形,侧棱SA⊥底面ABCD,AB垂直于AD和BC,SA ="AB=BC" =2,AD =1.M是棱SB的中点.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060157-54806.png)
(Ⅰ)求证:AM∥面SCD;
(Ⅱ)求面SCD与面SAB所成二面角的余弦值;
(Ⅲ)设点N是直线CD上的动点,MN与面SAB所成的角为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060157-97807.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060157-97807.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-39385.png)
(Ⅲ)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-97834.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-75727.png)
解析
试题分析:(Ⅰ)以点A为原点建立如图所示的空间直角坐标系,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-31438.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-39523.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-57790.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060159-18238.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060159-13176.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060159-17173.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060159-36737.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060159-69335.png)
设平面SCD的法向量是
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060200-34116.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060200-35236.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060200-38681.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060200-98625.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060201-22348.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060201-56504.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060201-91841.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060201-30809.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060201-50529.png)
(Ⅱ)易知平面SAB的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060202-36668.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060202-57370.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060202-29618.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060202-51748.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060201-50529.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-39385.png)
(Ⅲ)设
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060202-71021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060203-97061.png)
又,面SAB的法向量为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060202-36668.png)
所以,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060203-89175.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060203-33624.png)
当
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060203-98604.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-97834.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060158-75727.png)
点评:典型题,立体几何中平行、垂直关系的证明及角的计算问题是高考中的必考题,通过建立适当的坐标系,可使问题简化。
核心考点
试题【(本小题满分12分)如图,在四棱锥S - ABCD中,底面ABCD是直角梯形,侧棱SA⊥底面ABCD,AB垂直于AD和BC,SA ="AB=BC" =2,AD 】;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/20191022/20191022060140-69780.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060140-26983.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060140-44503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-72850.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060140-44503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-53868.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-94965.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-88574.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-43659.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-67781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-36623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-65545.png)
(I)当
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-67781.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-36623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-39007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060142-11751.png)
(II)要使二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060142-21389.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060142-89727.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060141-67781.png)
如图4,已知四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060108-65147.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-30372.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-43830.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-30372.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-28538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-25268.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-23117.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-89351.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060109-48953.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-78729.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-54423.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-52432.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-81255.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-54739.png)
(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-55094.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060110-21588.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060111-47800.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060101-76415.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060102-65216.png)
(1)求异面直线PA与CD所成的角;
(2)求证:PC∥平面EBD;
(3)求二面角A—BE--D的余弦值.
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060054-31886.jpg)
(1)求
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060054-71225.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060054-75128.png)
如图,在四棱锥
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060039-12342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-75738.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-58989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-75738.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-25813.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-38808.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-40794.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060041-60939.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060040-40794.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060041-10103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060041-84145.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060041-58391.png)
(1)求证:PD⊥平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060042-57991.png)
(2)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060042-86152.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060042-57991.png)
(3)求点
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060041-60939.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022060042-57991.png)
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