名校
1 . 如图所示,四棱锥
中,底面
为矩形,
平面
,
,
,点
是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/25/27e2bb49-56cf-4035-8d67-71418900b692.png?resizew=163)
(1)证明:
;
(2)求二面角
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83a04565a8ebaa111894b724b0ba266.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a88c44f558705de3bcefcfc0ece96b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/25/27e2bb49-56cf-4035-8d67-71418900b692.png?resizew=163)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4486d52b6e410fd7b60428121d96cef.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbeab80976db9b4689b9446cda06196a.png)
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解题方法
2 . 如图①所示,长方形
中,
,点M是边CD的中点,将
沿AM翻折到
,连结PB,PC,得到图②的四棱锥
.
(1)若棱PB的中点为N,求CN的长;
(2)设
的大小为
,若
,求平面
和平面
夹角余弦值的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8716b5aad93d97ca1c3791b9c717cc0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7f9fba8a4098c1a0515286eb8d616dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb62dd4766d11cfec3aee092b99e40c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfb216a6b4dfa1280e1e67d172fea409.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/26/6be74261-5155-450a-8e60-6f5c6bd730ff.png?resizew=374)
(1)若棱PB的中点为N,求CN的长;
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212e8c352c4d9b022a057d7d7fa7dd14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd571fb420cd6b9ecb83989f1d39c38f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c745df4f226027778d5fe45b6501b822.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
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3 . 如图1,平面四边形
中,
,
,
,E为
的中点,将
沿对角线
折起,使
,连接
,得到如图2所示的三棱锥
.
(1)证明:平面
平面
;
(2)已知直线
与平面
所成的角为
,求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98e486a1aad96167ff62f6fb5136e0bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bf10d92f20501e19d25f6f4159aab89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aa1162d5481e2441fe5bc0d49a576b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4357d5744046d4d44abb09e1ee35fcb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/26/8d6286fc-3716-438a-b431-6dfbbd9d7045.png?resizew=373)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d28c625d7ac6878957facc8274d459c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
(2)已知直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dfaad4c4467e27421876d8f2a4371d2.png)
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解题方法
4 . 如图,在三棱锥P-ABC中,∠ACB=90°,PA⊥底面ABC.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/1/9757cabc-de2f-4346-95e8-552c89440e03.png?resizew=163)
(1)求证:平面PAC⊥平面PBC;
(2)若AC=BC=PA,求平面PAB与平面PCB所成二面角的大小.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/1/9757cabc-de2f-4346-95e8-552c89440e03.png?resizew=163)
(1)求证:平面PAC⊥平面PBC;
(2)若AC=BC=PA,求平面PAB与平面PCB所成二面角的大小.
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2023-03-31更新
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682次组卷
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5卷引用:湖南省长沙市芙蓉高级中学2022-2023学年高二上学期期中数学试题
名校
解题方法
5 . 截角四面体是一种半正八面体,可由四面体经过适当的截角,即截去四面体的四个顶点所产生的多面体.如图所示,将棱长为
的正四面体沿棱的三等分点作平行于底面的截面,得到所有棱长均为a的截角四面体,则下列说法错误的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/c9279c45-53e4-4fde-8e18-d1c8022ffd15.png?resizew=165)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9878a063abcb6098d10560f2bf2d4b71.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/c9279c45-53e4-4fde-8e18-d1c8022ffd15.png?resizew=165)
A.二面角![]() ![]() |
B.该截角四面体的体积为![]() |
C.该截角四面体的外接球表面积为![]() |
D.该截角四面体的表面积为![]() |
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2023-01-12更新
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1396次组卷
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6卷引用:湖南省长沙市宁乡市第一高级中学2022-2023学年高三上学期12月月考数学试题
湖南省长沙市宁乡市第一高级中学2022-2023学年高三上学期12月月考数学试题重庆市第八中学校2022届高三下学期调研检测(十四)数学试题湖南省邵阳市2023届高三上学期一模数学试题(已下线)模块五 空间向量与立体几何-2(已下线)专题2 求二面角的夹角(2)专题15空间向量与立体几何(选填题)(2)
6 . 如图:直三棱柱
中,侧面
,
均为边长为2的正方形,且面
面
分别为正方形对角线
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/13/3401c7cc-8515-4505-9723-71b144e1ce36.png?resizew=171)
(1)求点
到面
的距离;
(2)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02cfa6b4db3a67fcd3c169fd8502a66d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cf9a6db3571fa57bfa2d5e4d44c51b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c49f0af09be9b2d3112b5bdb8b27f128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3db3b70cd3a7b12306eb4fe39a208b3d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/13/3401c7cc-8515-4505-9723-71b144e1ce36.png?resizew=171)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b948d211b0815c5ae923a458d6e4ec.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82cb18c10820d927ecd53326f58aaf8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ee5a94f9063a71581f409e47ebaf602.png)
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解题方法
7 . 如图,在三棱台
中,三棱锥
的体积为
,
的面积为
,
,且
平面
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/6/501cb360-14a2-4ca1-876a-2617f1f30ee1.png?resizew=211)
(1)求点
到平面
的距离;
(2)若
,且平面
平面
, 求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ac4fb99967c46a3855bcf2885b448c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3531142aafad00b62ad123b2646373e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c57c514077b8f020672946c22edfabcd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5f65f1481a9500babf018129a1d5124.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/458b6c1f8e142098dacb00c24c76aeb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c3ba78b8fe40bc19e33fda8ba8ffba7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d27df392ec1ca6478e552696fc43924.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/6/501cb360-14a2-4ca1-876a-2617f1f30ee1.png?resizew=211)
(1)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13b3b1cac8011583d3f5fe5d6eaa4a17.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b5648e9c0b455b35f7f997835aafa1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e3a59d7bf91a7540e35ce0011ad9b97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee3eb81e99c84589a3387d6ba0e6305a.png)
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2022-11-04更新
|
1968次组卷
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6卷引用:湖南省长沙市明达中学2022-2023学年高三上学期12月月考数学试题
名校
8 . 如图1,在平行四边形ABCD中,
,
,
,将△ABD沿BD折起,使得平面
平面
,如图2.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/15/fa496bfd-a347-40a2-9efe-ce372a5dae34.png?resizew=312)
(1)证明:
平面BCD;
(2)在线段
上是否存在点M,使得二面角
的大小为45°?若存在,求出
的值;若不存在,说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b8b98b2f83279a49e94d9f48c5e6f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e73bab2f24bffb9a51dab7cb1ee820f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f93edbd735d79524f463085a4e9093bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53d138354c4e021ac8ae2a2fb176ca14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fd148d264bc9043396f777523e907aa.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/10/15/fa496bfd-a347-40a2-9efe-ce372a5dae34.png?resizew=312)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d88e6824a54137282693946e80b57be9.png)
(2)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eb97aff0960e2640314888a38e7169c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64785e4401e1d79632e360fd3626ed62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f64221d0d4570490ae709f233242a3c.png)
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2022-10-12更新
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548次组卷
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4卷引用:湖南省长沙市雅礼中学2022-2023学年高二上学期第一次月考数学试题
名校
9 . 在正方体
中,下列几种说法正确的有( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
A.![]() | B.![]() |
C.![]() ![]() ![]() | D.二面角![]() ![]() |
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2022-10-07更新
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745次组卷
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4卷引用:湖南省长沙市第一中学2022-2023学年高三上学期月考(二)数学试题
湖南省长沙市第一中学2022-2023学年高三上学期月考(二)数学试题广东省深圳市罗湖外国语学校2023届高三上学期10月月考数学试题第十一章 立体几何初步 单元测试(已下线)第四章 立体几何解题通法 专题一 反证法 微点3 立体几何中的反证法综合训练【培优版】
10 . 蜂巢是由工蜂分泌蜂蜡建成的,从正面看,蜂巢口是由许多正六边形的中空柱状体连接而成,中空柱状体的底部是由三个全等的菱形面构成,菱形的一个角度是
,这样的设计含有深刻的数学原理.我著名数学家华罗庚曾专门研究蜂巢的结构,著有《谈谈与蜂房结构有关的数学问题》一书.用数学的眼光去看蜂巢的结构,如图,在六棱柱
的三个顶点
处分别用平面
,平面
,平面
截掉三个相等的三棱锥
,平面
,平面
,平面
交于点
,就形成了蜂巢的结构.如图,设平面
与正六边形底面所成的二面角的大小为
,则( )
![](https://img.xkw.com/dksih/QBM/2022/8/29/3054869176623104/3058462975041536/STEM/7d4c11340f694afcb2ae3c8ad08982c6.png?resizew=493)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/001640c8cf900976c8a973fb04b64c9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0395bb06ff1e38eaf3e5f7a5a790b269.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/feb0e488fc13f6fa31bdb241be399cad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13a1be205bf5955cb569d5eabde0eebb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e648c9ff4284df8551f924e34e00c131.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6895da13331cb525f5850d7b7a02a847.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8fbeb63b632886cc5eb62fb57d7032.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13a1be205bf5955cb569d5eabde0eebb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e648c9ff4284df8551f924e34e00c131.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6895da13331cb525f5850d7b7a02a847.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/347337898d085eeaf541523302700271.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://img.xkw.com/dksih/QBM/2022/8/29/3054869176623104/3058462975041536/STEM/7d4c11340f694afcb2ae3c8ad08982c6.png?resizew=493)
A.![]() | B.![]() |
C.![]() | D.![]() |
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2022-09-03更新
|
500次组卷
|
2卷引用:湖南省长沙市第一中学2022-2023学年高三上学期月考(一)数学试题