1 . 已知
平面
,
,
,
,
为
中点,过点
分别作平行于平面
的直线交
、
于点
、
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/17/96a2e6e5-9ef3-4b71-8b83-374077f21ea7.png?resizew=166)
(1)求直线
与平面
所成的角;
(2)证明:平面
平面
,并求直线
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91708c4508371f08556e76e31c7cb9ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a7b5adfcac0f46a4cd19da4ebb4a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/17/96a2e6e5-9ef3-4b71-8b83-374077f21ea7.png?resizew=166)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d48596215640e84a43fdd2a2c3a148d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ce6c0e9de83f2e64ae33609fc08459d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
您最近一年使用:0次
2 . 如图,在直三棱柱
中,
,
,且
分别是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/4fc86993-bed1-4ef5-b152-6c9ebc06d198.png?resizew=138)
(1)证明:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
;
(2)求三棱锥
的体积;
(3)求直线
与平面
所成角的大小.(结果用反三角函数值表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb4957406b21df59fdf7fa184752287b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c9d5815dc775d5a5810fff0b016a8d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3a51949f48ee8cf746851ba779b078e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/425f9b589e52ea2170ce988c4348b0eb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/4fc86993-bed1-4ef5-b152-6c9ebc06d198.png?resizew=138)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2daa808ca8c95f282dae5e1d578cb65.png)
(3)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
您最近一年使用:0次
名校
解题方法
3 . 高三新教学楼启用后,从一些教室窗口就能看到殷高路对面居民房平改坡后的屋顶(如图).其中
是屋脊线,
是屋檐线,
是屋顶坡面,
是一个与水平面垂直的带气窗的竖直面,
是气窗屋顶的屋脊线且
与竖直面
垂直.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/16/3cacf3fc-d38d-4a25-9bc2-80a2a7d0347d.png?resizew=252)
小张和小王对屋顶进行研究,提出了下面一些假设:
①两条屋脊线
与
互相垂直且都与水平面平行;
②气窗屋顶的两个坡面
与
互相垂直且与水平面的所成角相等;
③屋顶坡面
与水平面所成角为
.
(1)小张认为还需假设屋脊线
与带气窗的竖直面
是平行关系.而小李认为前面的假设已经够了,不需要再提出这个假设.请你判断哪位同学正确?证明你的判断.
(2)根据小张和小王的假设,试求气窗屋顶的一个坡面
与屋顶坡面
构成的阴脊线
(是平面
与平面
的交线)与水平面所成角的大小.(用反三角函数表示)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02b1139e07e431b5d4276757b232bad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/16/3cacf3fc-d38d-4a25-9bc2-80a2a7d0347d.png?resizew=252)
小张和小王对屋顶进行研究,提出了下面一些假设:
①两条屋脊线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
②气窗屋顶的两个坡面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a681d311a864d38cf306a0c137cbcca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f515246d9ee093bfbf9983fd47357d.png)
③屋顶坡面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02b1139e07e431b5d4276757b232bad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ac09dc1ca2cdd7aef28c218763d3e4d.png)
(1)小张认为还需假设屋脊线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
(2)根据小张和小王的假设,试求气窗屋顶的一个坡面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2257da1e2425f2ea9ac7440f52659ff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02b1139e07e431b5d4276757b232bad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02b1139e07e431b5d4276757b232bad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f515246d9ee093bfbf9983fd47357d.png)
您最近一年使用:0次
4 . 如图,在四棱锥
中,
⊥平面
,正方形
的边长为
,
,设
为侧棱
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/3b4b1972-387a-4e1f-ba7f-9019f00d6f13.png?resizew=146)
(1)求四棱锥
的体积
;
(2)求直线
与平面
所成角
的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00bab2c27eac56fffa4cd7dbe1dcdf1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/24/3b4b1972-387a-4e1f-ba7f-9019f00d6f13.png?resizew=146)
(1)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
您最近一年使用:0次
2022-11-23更新
|
537次组卷
|
8卷引用:上海市宝山区高境一中2018-2019学年高二下学期期中数学试题
名校
5 . 如图,在直三棱柱
中,已知
,
,
.
的体积;
(2)求直线
与平面
所成的角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/209acf15985d1ea1ad86fc4a37e38c0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cafb1f6c1f0856b4961eecaf49731eb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080ca48cd27d4bf9d9ef084b558fc17a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2d0727de4c16b53b4bb6ab370afde6c.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
您最近一年使用:0次
2022-05-08更新
|
586次组卷
|
5卷引用:上海市行知中学2021-2022学年高二下学期期中数学试题
上海市行知中学2021-2022学年高二下学期期中数学试题(已下线)第11讲 柱、锥、台的体积(核心考点讲与练)-2022-2023学年高二数学考试满分全攻略(沪教版2020必修第三册)(已下线)11.2锥体(作业)(夯实基础+能力提升)-【教材配套课件+作业】2022-2023学年高二数学精品教学课件(沪教版2020必修第三册)(已下线)核心考点05 空间向量及其应用(2)(已下线)11.2 锥体(第2课时)(三大题型)(分层练习)-2023-2024学年高二数学同步精品课堂(沪教版2020必修第三册)
名校
6 . 如图,正方形
的边长为2,E,F分别是边
及
的中点,将
,
及
折起,使A、C、B点重合于
点.
![](https://img.xkw.com/dksih/QBM/2022/4/26/2966573015228416/2967895166992384/STEM/286cd9c4-9973-4925-bb9c-ae665421f956.png?resizew=156)
(1)求三棱锥
的体积;
(2)求
与平面
所成角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c771a4feb150ad9cff8d70431c97eb17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f2ea13010e2399194be2a681310543e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13668f033d00acfc366f7e47949c4462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://img.xkw.com/dksih/QBM/2022/4/26/2966573015228416/2967895166992384/STEM/286cd9c4-9973-4925-bb9c-ae665421f956.png?resizew=156)
(1)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce074ab853855f689812c161f3160ac9.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134ef0b1a2669a09f05bd4dc2496f706.png)
您最近一年使用:0次
名校
解题方法
7 . 如图,已知PA=AC=PC=AB=a,
,
,M为AC的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/13/ae110279-8d2e-4a9a-9d8b-b6397c51dee3.png?resizew=143)
(1)求证:
平面ABC;
(2)求直线PB与平面ABC所成角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbb05b8b630052ff544249ebd72d95d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/822ba132ca9dd0d4a050659aef3c9b26.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/13/ae110279-8d2e-4a9a-9d8b-b6397c51dee3.png?resizew=143)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a7442b64b37f685bc3ae88ff450c1a8.png)
(2)求直线PB与平面ABC所成角的大小.
您最近一年使用:0次
2022-04-23更新
|
349次组卷
|
4卷引用:上海市宝山区海滨中学2023-2024学年高二上学期10月学业质量检测数学试题
上海市宝山区海滨中学2023-2024学年高二上学期10月学业质量检测数学试题上海市虹口区2018届高三上学期期末教学质量监控数学试题沪教版(2020) 必修第三册 同步跟踪练习 第10章 10.3.3 直线与平面所成的角(已下线)上海市华东师范大学第二附属中学2022-2023学年高二下学期开学考试数学试题
8 . 如图,在四棱锥P-ABCD中,已知PA⊥平面ABCD,且四边形ABCD为直角梯形,∠ABC=∠BAD=90°,AB=AD=AP=2,BC=1,且Q为线段BP的中点.
![](https://img.xkw.com/dksih/QBM/2022/3/9/2932303732539392/2933791275737088/STEM/75e64e92c2114d388dd8c939335e6601.png?resizew=174)
(1)求直线CQ与PD所成角的大小;
(2)求直线CQ到平面ADQ所成角的大小.
![](https://img.xkw.com/dksih/QBM/2022/3/9/2932303732539392/2933791275737088/STEM/75e64e92c2114d388dd8c939335e6601.png?resizew=174)
(1)求直线CQ与PD所成角的大小;
(2)求直线CQ到平面ADQ所成角的大小.
您最近一年使用:0次
2022-03-11更新
|
394次组卷
|
4卷引用:上海市交通大学附属中学2022届高三下学期开学考数学试题
上海市交通大学附属中学2022届高三下学期开学考数学试题上海市嘉定区2021届高三三模数学试题(已下线)考向24空间向量与立体几何-备战2022年高考数学一轮复习考点微专题(上海专用)上海市2023届高三下学期开学摸底数学试题
9 . 如图,长方体中
中,
,点P为面
的对角线
上的动点(不包括端点),PN⊥BD于N.
![](https://img.xkw.com/dksih/QBM/2021/10/19/2832962918006784/2833684082114560/STEM/695b8c39fe9644769b9c4cdbbcf554c0.png?resizew=209)
(1)若点P是
的中点,求线段PN的长度;
(2)设
,将PN表示为
的函数,并写出定义域;
(3)当PN最小时,求直线PN与平面ABCD所成角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b98a13eacfcc6743aa433d7674e18e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ebb05874eb3353d754af24c9974273e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c09eec4e14a861af83d7828797d176.png)
![](https://img.xkw.com/dksih/QBM/2021/10/19/2832962918006784/2833684082114560/STEM/695b8c39fe9644769b9c4cdbbcf554c0.png?resizew=209)
(1)若点P是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c09eec4e14a861af83d7828797d176.png)
(2)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8711eddf26d11fc974dfb6da4b640918.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
(3)当PN最小时,求直线PN与平面ABCD所成角的大小.
您最近一年使用:0次
2021-10-20更新
|
272次组卷
|
5卷引用:上海市宝山中学2021-2022学年高二上学期10月月考数学试题
上海市宝山中学2021-2022学年高二上学期10月月考数学试题上海师范大学第二附属中学2021-2022学年高二上学期期中数学试题上海市松江区第四中学2022-2023学年高二上学期期中数学试题(已下线)10.3 直线与平面所成的角 (第4课时)(作业)(夯实基础+能力提升)-【教材配套课件+作业】2022-2023学年高二数学精品教学课件(沪教版2020必修第三册)(已下线)期中真题必刷压轴60题(18个考点专练)-【满分全攻略】2023-2024学年高二数学同步讲义全优学案(人教A版2019选择性必修第一册)
名校
解题方法
10 . 如图,在四棱锥
中,底面
为正方形,
底面
,
,E为线段
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/11ebdb62-a2e3-4a92-8747-fe1a5f84c6fb.png?resizew=168)
(1)若F为线段
的中点,求直线
和平面
所成角的大小.
(2)若点F在线段
上移动,当三棱锥
体积最大时,求异面直线
与
所成角的大小.
![](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/829f9180ddd9aa1a0ee0dc520f4e0b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/11ebdb62-a2e3-4a92-8747-fe1a5f84c6fb.png?resizew=168)
(1)若F为线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若点F在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccc3bf74119692ac98eb24fcfa2a3f9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
您最近一年使用:0次
2021-10-18更新
|
308次组卷
|
3卷引用:上海市行知中学2021-2022学年高二上学期10月月考数学试题
上海市行知中学2021-2022学年高二上学期10月月考数学试题上海市格致中学2022届高三上学期12月月考数学试题(已下线)11.2锥体(作业)(夯实基础+能力提升)-【教材配套课件+作业】2022-2023学年高二数学精品教学课件(沪教版2020必修第三册)