题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235232-58590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235232-52501.png)
(1)求证:AD^BC
(2)求二面角B-AC-D的大小
(3)在直线AC上是否存在一点E,使ED与面BCD成30°角?若存在,确定E的位置;若
不存在,说明理由.
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-43363.png)
(3)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-18298.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-74981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-56280.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-71606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-97591.png)
解析
(1)利用线面垂直的性质定理得到证明。
(2)合理的建立空间直角坐标系,表示平面的法向量,借助于向量的数量积的性质定理,表示法向量的夹角,得到二面角的平面角的大小。
(3)对于探索性问题,可以假设存在,然后在此基础上,我们进一步分析斜向量和平面的法向量,利用线面角的大小求解得到。
解: (1)方法一:作
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-37325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235235-29563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235235-65614.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235235-20527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235235-48383.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235236-24562.png)
又
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235236-56139.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235236-70605.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235236-79577.png)
方法二:取
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235237-73191.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235237-65067.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235237-82221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235237-73388.png)
则有
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235237-80560.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235238-96711.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235238-13813.png)
(2)作
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235238-36915.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235239-50227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235239-45421.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235239-32359.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235239-50003.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235239-66984.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235240-80949.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235240-92505.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235239-50227.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-18298.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235240-10347.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235240-89247.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235241-67486.png)
由余弦定理得
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235241-56628.png)
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-74981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235241-60854.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235241-71904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235242-54219.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235242-88180.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235242-91467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235242-19834.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235243-98707.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-71606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235243-39851.png)
设
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235243-34396.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235243-54847.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235243-10456.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235244-13973.png)
故线段
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-18298.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-74981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-56280.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-71606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-97591.png)
解法二:
(1)作
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-37325.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235235-29563.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235244-28038.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235244-65524.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235244-86195.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235236-70605.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235245-91427.png)
以
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235245-46085.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235245-82906.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235245-95240.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235245-73000.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235246-33134.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235246-40662.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235246-95359.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235246-73138.png)
(2)设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235247-97647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235247-76308.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235247-65643.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235247-97995.png)
同理由
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235248-65556.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235248-33129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235248-54623.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235248-26410.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235248-95033.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235240-80949.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235249-98843.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235249-14590.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235249-98843.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235249-86501.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-43363.png)
(3)设
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235249-18677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-18298.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235250-13298.png)
平面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235250-26353.png)
要使
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-71606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-97591.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235250-43073.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235250-58145.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235250-26810.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235251-47047.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235251-76184.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235251-82334.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235251-33242.png)
故线段
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-18298.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-74981.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235233-56280.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-71606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-14417.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235234-97591.png)
核心考点
试题【如图,在三棱锥A-BCD中,侧面ABD、ACD是全等的直角三角形,AD是公共的斜边,且AD=,BD=CD=1,另一个侧面是正三角形(1)求证:AD^BC(2)求】;主要考察你对空间几何体的结构特征等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235226-83503.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235226-26384.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235226-79606.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235226-62928.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235226-62492.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235227-71003.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235227-32643.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235227-37779.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235227-30099.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235228-87135.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235219-48199.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235219-40072.png)
① 若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235219-25530.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235220-90692.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235220-22869.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235220-38797.png)
③ 若
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235220-29430.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235220-99234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235221-55264.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235221-75992.png)
其中正确的命题是 .(写出所有正确命题的序号)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235213-87557.png)
(1)求证:EF⊥CF;
(2)求EF与CG所成的角的余弦值;
(3)求三棱锥G-CEF的体积.
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235205-99163.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235206-75597.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235206-51990.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235206-87566.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235158-90739.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235158-37989.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235158-26930.jpg)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235159-10884.png)
(2)求多面体
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235159-88473.png)
(3)试问:在线段AB上是否存在一点N,使面
![](http://img.shitiku.com.cn/uploads/allimg/20191020/20191020235159-98420.png)
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