在三棱锥
中,顶点P在平面ABC内的射影是
的外心,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30fc65a72853bd8ac1ad0828270d3baf.png)
20-21高一·全国·课后作业 查看更多[2]
更新时间:2021-11-13 10:09:59
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【知识点】 线面垂直证明线线垂直
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解答题-问答题
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【推荐1】如图(1),平面四边形
中,
,
,
,将
沿
边折起如图(2),使
,点
,
分别为
,
中点.
![](https://img.xkw.com/dksih/QBM/2022/5/24/2986198871629824/2986300714467328/STEM/9bc11e39-d190-43fc-89e8-16549b951975.png?resizew=520)
(1)判断直线
与平面
的位置关系,并说明理由;
(2)求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/520f8abda6a85e7ef6f281fc2df853fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b81939c1f23fa5fb48a3a270bbf52d60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121eabff3c62c1a196d9ca5f6f83f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037b342a682cbd4241855a243da3c016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7650cede07c4758a9b3bb1da4553acc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://img.xkw.com/dksih/QBM/2022/5/24/2986198871629824/2986300714467328/STEM/9bc11e39-d190-43fc-89e8-16549b951975.png?resizew=520)
(1)判断直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0912d666aa93db05c94bb8c0368a9790.png)
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【推荐2】如图.在四棱锥P-ABCD中,
平面ABCD,且![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55babe05bc255e22fa009c55c4d4446.png)
,
.
(2)求点A到平面PCD的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55babe05bc255e22fa009c55c4d4446.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25ab46164b23af7a4c4907f176e392ec.png)
(2)求点A到平面PCD的距离.
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解题方法
【推荐3】在我国古代数学名著《九章算术》中将由四个直角三角形组成的四面体称为“鳖臑”.已知三棱锥P-ABC中,PA⊥平面ABC.
(2)已知三棱锥P-ABC是一个“鳖臑”,且AC=1,AB=2,∠BAC=60°.
①若△PAC上有一点D,如图1所示,试在平面PAC内作出一条过点D的直线l,使得l与BD垂直,说明作法,并给予证明;
②若点D在线段PC上,点E在线段PB上,如图2所示,且PB⊥平面EDA,证明∠EAB是平面EAD与平面BAC的二面角的平面角.
(2)已知三棱锥P-ABC是一个“鳖臑”,且AC=1,AB=2,∠BAC=60°.
①若△PAC上有一点D,如图1所示,试在平面PAC内作出一条过点D的直线l,使得l与BD垂直,说明作法,并给予证明;
②若点D在线段PC上,点E在线段PB上,如图2所示,且PB⊥平面EDA,证明∠EAB是平面EAD与平面BAC的二面角的平面角.
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