题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-65641.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-10133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-10133.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-31366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-69508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-57218.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-78507.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-69538.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-78398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-75975.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-31366.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-38985.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-11246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-25937.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-97809.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-61116.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
(2)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-62346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96013.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96268.png)
解析
试题分析:(1)平行关系的证明问题问题,要注意三角形中位线定理的应用,注意平行关系的传递性,以及线线关系、线面关系、面面关系的相互转化;
(2)立体几何中的求角问题,往往有两种思路,即“几何法”和“向量法”.本题应用“几何法”,应注意“一作,二证,三计算”,注意在直角三角形中解决问题;
应用“向量法”,要注意利用已有的垂直关系,一建立空间直角坐标系.
本题建系后,确定点的坐标及平面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-56974.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-36863.png)
计算得到
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-82694.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-62346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96268.png)
试题解析:(1)连结
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-15123.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-83732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-77303.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-77303.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-83732.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-75975.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-38985.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-52735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-34150.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-54597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-52735.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
连结
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-34342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-34342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-62346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-53095.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-34342.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-38894.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-77259.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-18736.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-39028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-10374.png)
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-98625.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-78244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-86246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-77133.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-47783.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-86246.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113249-54597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-64644.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-53108.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-76769.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-65891.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113246-69538.png)
由余弦定理知
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-69046.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113250-74578.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113251-27007.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113251-69421.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113251-36924.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113251-23020.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113251-65689.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-62346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96013.png)
在直角三角形
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113251-83898.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113252-49884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113252-48383.png)
法二:以
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-78398.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113252-89022.png)
设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-56974.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113253-33617.png)
由
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113253-25687.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113253-41757.png)
令
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113253-57767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113253-70285.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113254-66100.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-36863.png)
∴
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-82694.png)
∴直线
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113247-62346.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96013.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113248-96268.png)
核心考点
试题【如图,矩形所在的平面和平面互相垂直,等腰梯形中,∥,=2,,,,分别为,的中点,为底面的重心.(1)求证:∥平面;(2)求直线与平面所成角的正弦值.】;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/20191105/20191105113228-36254.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113229-36406.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113229-87758.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113229-73036.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113229-39314.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113230-24412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113230-81079.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113230-63019.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113230-75103.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113231-89718.png)
(1)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113231-94412.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113231-77413.png)
(2)求面
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113231-21636.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113231-77413.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113216-77451.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113216-49418.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113216-30967.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113216-90112.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113216-20183.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113217-80028.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113217-59244.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113217-11921.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113217-70349.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113217-12393.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113217-83767.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113209-87297.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113209-96794.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113205-18734.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113159-58609.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113159-12756.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113159-34093.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113159-22384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191105/20191105113200-35572.jpg)
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