解题方法
1 . 如图①所示,在
中,
分别是棱
和
的中点.如图②所示,现沿
将
折起到
的位置,使平面
底面
,过点
作
于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/28/b78f85a2-a3b6-4e50-acfe-d06b9ccd6b7b.png?resizew=327)
(1)求证:
平面
;
(2)求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77504b6aead2e3715ace061124e708f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cff7399ecc698e2fb415147c96d0d03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac48b9ac8efbf41d6ab5242d247bd89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3a079cfdcca9acdacecbf08f9f78cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7459863f058993e17b7dcf902053eccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/28/b78f85a2-a3b6-4e50-acfe-d06b9ccd6b7b.png?resizew=327)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36fd25c7dcc396c237164cbc9891605b.png)
您最近一年使用:0次
解题方法
2 . 如图,菱形
中,
,
,
为
上一点,满足
,将菱形沿
对折,形成四面体
,满足![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/28/cbac913e-1fbd-44d0-b009-cac65bb07b26.png?resizew=346)
(1)设折叠前
的面积为
,折叠后的面积为
,求
的值;
(2)求三棱锥
的体积
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e13c772461aef1d9d715129636739748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9244463ccb95a97d1282e427d54886e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3931333820859378ea6723ff3075189.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/28/cbac913e-1fbd-44d0-b009-cac65bb07b26.png?resizew=346)
(1)设折叠前
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c771a4feb150ad9cff8d70431c97eb17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e097c8d4c948de063796bd19f85b3a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e0bd63f55069a3bc870915010b39225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/235f0a6fb218d28383e6f27f2df1f50f.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52753d89bf58589e2e83b19bd3d140b8.png)
您最近一年使用:0次
2022-07-15更新
|
501次组卷
|
6卷引用:贵州省遵义市2021-2022学年高二下学期期末质量监测数学(文)试题
贵州省遵义市2021-2022学年高二下学期期末质量监测数学(文)试题河北省衡水市阳光中学2022-2023学年高二上学期开学考试数学试题(已下线)8.3.1棱柱、棱锥、棱台的表面积和体积(精练)-【精讲精练】2022-2023学年高一数学下学期同步精讲精练(人教A版2019必修第二册)(已下线)8.3.1 棱柱、棱锥、棱台的表面积和体积(2)-2022-2023学年高一数学《考点·题型·技巧》精讲与精练高分突破系列(人教A版2019必修第二册)(已下线)立体几何专题:折叠问题中的证明与计算5种题型(已下线)微专题12 轻松搞定空间几何体的体积问题(2)
解题方法
3 . 如图,菱形ABCD中,
,
,E为BC上一点,满足
,将棱形沿BD对折,形成四面体C-ABD,满足
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/22/196d092d-b285-4dcb-a84d-6da84751bc92.png?resizew=303)
(1)求三棱锥E-ABD的体积;
(2)求二面角A-DE-B的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e13c772461aef1d9d715129636739748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e73c8c1d2ba6b29b301380a45dfbcdd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/22/196d092d-b285-4dcb-a84d-6da84751bc92.png?resizew=303)
(1)求三棱锥E-ABD的体积;
(2)求二面角A-DE-B的正弦值.
您最近一年使用:0次
名校
解题方法
4 . 如图,在四棱锥
中,
,
,
,
,
,
,
都在平面
的上方.
![](https://img.xkw.com/dksih/QBM/2022/7/13/3021598195507200/3023065516761088/STEM/c7774c2fdf0a497f81749b448c5c6aa1.png?resizew=166)
(1)证明:平面
平面
;
(2)若
,且平面CDE与平面ABE所成锐二面角的余弦值为
,求四棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d730ae4307db56b47849c3a19dedfb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/829018a6ca0aff95d89e3f7cd943274e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50f218914337edd06e59e75d90b777e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63367b1d9c4f1c3e989ed5d881d0e3e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
![](https://img.xkw.com/dksih/QBM/2022/7/13/3021598195507200/3023065516761088/STEM/c7774c2fdf0a497f81749b448c5c6aa1.png?resizew=166)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39282bdf319f30d7bc261e2e3ab3b1e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64b3539fcb35e07fcf3339eb04e7748d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/793b2e12599fadb08caf8642e000363e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
您最近一年使用:0次
2022-07-15更新
|
704次组卷
|
4卷引用:贵州省黔西南州2021-2022学年高二下学期期末质量检测数学(理)试题
贵州省黔西南州2021-2022学年高二下学期期末质量检测数学(理)试题山西省忻州市2021-2022学年高二下学期期末联合考试数学试题(已下线)第06讲 向量法求空间角(含探索性问题) (讲)-2江西省九校2024届新高三上学期联合考试数学试题
解题方法
5 . 如图,菱形
的边长为4,
,矩形
的面积为8,且平面
平面
.
![](https://img.xkw.com/dksih/QBM/2022/7/4/3015174232465408/3016233085861888/STEM/eeab24e8af0048f596ea38229facbd80.png?resizew=214)
(1)证明:
;
(2)求C到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea4f5eec0addba78f2e0cdfb7ecc59a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ac66de8543430fd51e7c18042e626dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed9c7b69462dc9226e35ddac8f179cf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://img.xkw.com/dksih/QBM/2022/7/4/3015174232465408/3016233085861888/STEM/eeab24e8af0048f596ea38229facbd80.png?resizew=214)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589ddae20626f9aaac616d2a3b5d95bd.png)
(2)求C到平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
您最近一年使用:0次
2022-07-05更新
|
112次组卷
|
2卷引用:贵州省六盘水第五中学2021-2022学年高二上学期期末数学试题
解题方法
6 . 如图,在正三棱柱
中,
,
,
,
分别为
,
,
的中点.
![](https://img.xkw.com/dksih/QBM/2022/2/7/2911409210753024/2947367577894912/STEM/4d99daa6-3636-4a1b-9289-9eb3d0b637c9.png?resizew=140)
(1)证明:
.
(2)求平面
与平面
所成锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92105835f8075cb75dff244e908370b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://img.xkw.com/dksih/QBM/2022/2/7/2911409210753024/2947367577894912/STEM/4d99daa6-3636-4a1b-9289-9eb3d0b637c9.png?resizew=140)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a7a3dc3f3a02f4400e22dec2f2fee23.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c0bfeadcf17b2a45896071f07a4a5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134ef0b1a2669a09f05bd4dc2496f706.png)
您最近一年使用:0次
2022-03-30更新
|
174次组卷
|
2卷引用:贵州省毕节市2021-2022学年高二上学期期末教学质量检测数学(理)试题
2013·山东·一模
名校
解题方法
7 . 如图所示,已知
平面ACD,
平面ACD,
为等边三角形,
,F为CD的中点.求证:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/28/35cb2e1c-0183-4e19-b23b-ee240a0a6992.png?resizew=169)
(1)
平面BCE;
(2)平面
平面CDE.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21f9157fce2a8339d281178c7c0bccbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8d2d217e9bcd059908f117dfc4d4259.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb59a3752da728cfa77557dd14d0f737.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/28/35cb2e1c-0183-4e19-b23b-ee240a0a6992.png?resizew=169)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c9c89e28bb3b5ce434e8ebea6363339.png)
(2)平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39282bdf319f30d7bc261e2e3ab3b1e4.png)
您最近一年使用:0次
2022-02-26更新
|
3584次组卷
|
27卷引用:贵州省黔西南州2021-2022学年高一下学期期末质量检测数学试题
贵州省黔西南州2021-2022学年高一下学期期末质量检测数学试题河北省枣强中学2016-2017学年高二下学期期末考试数学(文)试题西藏自治区拉萨中学2020-2021学年高一下学期期末考试数学试题(已下线)宁夏回族自治区石嘴山市第三中学2022-2023学年高一下学期期末考试数学试卷宁夏石嘴山市第三中学2022-2023学年高一下学期期末数学试题(已下线)2013届山东省高三高考压轴文科数学试卷(已下线)2015高考数学(理)一轮配套特训:7-5直线、平面垂直的判定及性质2016届河南郑州一中教育集团高三文押题二数学试卷2017届江西省鹰潭市高三第一次模拟考试数学(文)试卷2017届江西省南昌市十所省重点中学命制高三第二次模拟突破冲刺二数学(文)试卷江西省九江第一中学2016-2017学年高二下学期期中考试数学(文)试题(已下线)2017-2018学年第一学期期末复习备考之精准复习模拟题高一人教版(必修一+必修二)数学试题(B卷)山东省夏津一中2019届高三上学期12月月考数学(文)试题(已下线)7-5 直线、平面垂直的判定及其性质(高效训练)-2019版导学教程一轮复习数学(人教版)(已下线)江西省鹰潭市2017届高三第一次模拟考试文数试题(已下线)专题42 空间点、直线、平面的位置关系综合练习-2021年高考一轮数学(理)单元复习一遍过(已下线)专题42 空间点、直线、平面的位置关系综合练习-2021年高考一轮数学单元复习一遍过(新高考地区专用)(已下线)专题42 空间点、直线、平面的位置关系综合练习-2021年高考一轮数学(文)单元复习一遍过甘肃省金昌市永昌县第一高级中学2021-2022学年高三上学期12月月考数学文科试题河南濮阳市华龙区高级中学2021-2022学年高三上学期开学考试数学文科试题(已下线)专题8.2 立体几何初步 章末检测2(中)-【满分计划】2021-2022学年高一数学阶段性复习测试卷(人教A版2019必修第二册)(已下线)第八章 立体几何初步(基础训练)A卷-2021-2022学年高一数学课后培优练(人教A版2019必修第二册)吉林省长春市第二实验中学2022-2023学年高一下学期期中考试数学试题第十一章 立体几何初步 单元检测卷河南省南阳市卧龙区博雅学校2022-2023学年高一下学期6月月考数学试题四川省成都市锦江区嘉祥外国语高级中学2022-2023学年高一下学期6月月考数学试题广东省深圳市第二高级中学2023-2024学年高二上学期第一学段考试数学试题
名校
解题方法
8 . 如图,在四棱锥
中,底面
是矩形,
平面
,
为垂足.
在线段
上移动时,判断
是否为直角三角形,并说明理由;
(2)若
,当点
是
的中点,且
时,求三棱锥
的体积.
![](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/a1597af5a4405ce68f5a97c87de4df7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83a04565a8ebaa111894b724b0ba266.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d03c2639a3b3f1f9590080b38ab21374.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2b24136c688c1dcb489dd67da5154d3.png)
您最近一年使用:0次
2022-02-22更新
|
322次组卷
|
2卷引用:贵州省贵阳市普通中学2022届高三上学期期末监测考试数学(文)试题
名校
解题方法
9 . 在四棱锥
中,底面
是直角梯形,
,
,
,
分别是棱
,
的中点.
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899231840321536/2902340650344448/STEM/e33bb52e-b185-4986-8bd7-1801e37e071c.png?resizew=215)
(1)证明:
平面
;
(2)若
,且四棱锥
的体积是6,求三棱锥
的体积.
![](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/d730ae4307db56b47849c3a19dedfb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9060f03b9ee41d70d135b1e1a8902ce9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899231840321536/2902340650344448/STEM/e33bb52e-b185-4986-8bd7-1801e37e071c.png?resizew=215)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f9d682e5d3cc8573574d8d11636758.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e84094aedc798143d465276916c1b9a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f09d555c9022f7546fe4a678b599376.png)
您最近一年使用:0次
2022-01-25更新
|
515次组卷
|
7卷引用:贵州省名校联盟2022届高三上学期期末数学(文)试题
名校
解题方法
10 . 在四棱锥
中,
平面ABCD,底面ABCD是直角梯形,
,
,E,F分别是棱AB,PC的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/31/3435e5b1-4cfd-4efd-8ecf-358587ecbe07.png?resizew=203)
(1)证明:
平面PAD.
(2)若
,
,求平面AEF与平面CDF所成锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a755edadca4e4fc27fd49559b8d691ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9060f03b9ee41d70d135b1e1a8902ce9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/31/3435e5b1-4cfd-4efd-8ecf-358587ecbe07.png?resizew=203)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06222ee533c2484ab25321a6abbf98cb.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ace900749d0861aa51fcc6d72c51f82c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e839ac941e8bf536ff35a12e56c7a400.png)
您最近一年使用:0次
2022-01-24更新
|
516次组卷
|
5卷引用:贵州省名校联盟2022届高三上学期期末数学(理)试题