名校
解题方法
1 . 在正四棱柱
中,
为
中点,直线
与平面
交于点
.
为
的中点;
(2)若直线
与平面
所成的角为
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35d0134fcec6237e81fc52e289c2e564.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2331bccb6ebf5b9fd639df994f575a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
(2)若直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2331bccb6ebf5b9fd639df994f575a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1a63ab608517bb10aa036783dfb51f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f492d9fcc189382b109dc342e807d50.png)
您最近一年使用:0次
2024-04-24更新
|
892次组卷
|
2卷引用:北京市丰台区2023-2024学年高三下学期综合练习(二)数学试题
2 . 在三棱柱
中,
,平面
平面
,
分别为棱
的中点,如图:
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/21/cce4fc8f-3773-44a9-a5a8-8ca21ea6d283.png?resizew=181)
(1)求证:
平面
;
(2)若
,
①求
与平面
所成角的正弦值;
②求线段
在平面
内的投影
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5a1d8b135a43429bba122bb000ca83e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3e34cc1159ab9198480cd0b585620d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9369aed2d8309af46ac3eaffb9cce537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d172b97111c32fa11369a6c59719c8f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/21/cce4fc8f-3773-44a9-a5a8-8ca21ea6d283.png?resizew=181)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17e5e2ba78a5b1dd0f39bb65d2a0a0f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36c4559d27e3905980d1a4f1856f07de.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39bfd8e9f2f08a5807a23677988b240b.png)
②求线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39bfd8e9f2f08a5807a23677988b240b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fae626b30192bba5c433d399bba65411.png)
您最近一年使用:0次
2023-12-21更新
|
289次组卷
|
2卷引用:北京市西城区北师大附属实验中学2024届高三上学期12月月考数学试题
解题方法
3 . 如图,在四棱锥
中,底面ABCD为平行四边形,平面
平面ABCD,
,
,点M为棱PC中点,平面ABM与棱PD交于点N.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/27/943d64f7-4c44-4159-8227-e067f026977d.png?resizew=183)
(1)求证:N是棱PD的中点;
(2)再从条件①、条件②这两个条件中选择一个作为已知,求:
(i)二面角
的余弦值;
(ii)在棱PA上是否存在点Q,使得
平面BDM?若存在,求出
的值;若不存在,说明理由.
条件①:
;
条件②:
.
注:如果选择条件①和条件②分别解答,按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19ad6a0124359e8b9f7649cf0bff51ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37002ada5d194d4d062fa3285d7d9824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7d9f95acb654472e6a0b00554e5f7c4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/27/943d64f7-4c44-4159-8227-e067f026977d.png?resizew=183)
(1)求证:N是棱PD的中点;
(2)再从条件①、条件②这两个条件中选择一个作为已知,求:
(i)二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64785e4401e1d79632e360fd3626ed62.png)
(ii)在棱PA上是否存在点Q,使得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c5197adf1af97b29adc08417400807c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc953f0be1dafec1b4d1836cbafbf59.png)
条件①:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f945a69cf7e8213e50622125cde652f5.png)
条件②:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714cc3707bba3bfdb56e251999be8592.png)
注:如果选择条件①和条件②分别解答,按第一个解答计分.
您最近一年使用:0次
名校
4 . 如图,在五面体ABCDEF中,四边形ABCD是矩形,
.
(1)求证:
:
(2)从下面三个条件中选择一个作为已知,使五面体ABCDEF存在.求直线AE与平面BCF所成角的正弦值.
条件①:平面
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件②:平面
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件③:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7dac702fe64edf1bc265da4b98cf2a0.png)
注:如果选择的条件不符合要求,得0分;如果选择多个条件分别解答,按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04dee62ab0a09454e3906fbd0d3f14a0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/12/af0a558d-cb99-475b-9565-df5ed4ffd77f.png?resizew=200)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/333ab24c4935210f4c232cd0c0fae358.png)
(2)从下面三个条件中选择一个作为已知,使五面体ABCDEF存在.求直线AE与平面BCF所成角的正弦值.
条件①:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb5a68a008a22d5a8cea5fe8dcf31e10.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件②:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d28c625d7ac6878957facc8274d459c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件③:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7dac702fe64edf1bc265da4b98cf2a0.png)
注:如果选择的条件不符合要求,得0分;如果选择多个条件分别解答,按第一个解答计分.
您最近一年使用:0次
名校
5 . 如图,在长方体
中,侧面
是正方形,且
,点E为BC的中点,点F在直线
上.
(1)若
平面
,求证:
平面
;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab3e0dba5705e1d749cfb21ebbb2ed93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47f4f0f539f5b418e4c8169b72f96067.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/30/6e793309-44da-4ae7-a663-a7064c18efd5.png?resizew=168)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0447e46f2d9b39960ae1f1294ed8a2f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8464f8d92d471c5827bf8c94b6ea12db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94270844f197d524bf1da4f1385befd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8464f8d92d471c5827bf8c94b6ea12db.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b45a381a33453534652f66c351379967.png)
您最近一年使用:0次
2023-08-30更新
|
503次组卷
|
3卷引用:北京市2024届新高三入学定位考试数学试题
北京市2024届新高三入学定位考试数学试题北京市东直门中学2024届高三上学期开学考试数学试题(已下线)专题06 二面角4种常见考法归类 - 【考点通关】2023-2024学年高二数学高频考点与解题策略(人教B版2019选择性必修第一册)
6 . 如图,在三棱锥
中,
,
,
,
.
平面
;
(2)若
,
,求二面角
的平面角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c646c683fbe522edb7ea54fd3ad873d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebc51db970e3290c359881c7924a1c16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bcbf1197b6a9e629dbd76ba6b8fbd81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b80ee363635d73f601654339028daec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89b336e518ac4ff04c6c26e4b8a15844.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47d294d69caac577339f11f477b2047e.png)
您最近一年使用:0次
2024-04-18更新
|
1149次组卷
|
2卷引用:北京市通州区潞河中学2023-2024学年高三下学期第三次模拟数学试卷
名校
解题方法
7 . 如图,在四棱锥
中,底面
为矩形,平面
平面
,
,
,
,
,
分别是
,
的中点.
(1)求证:
平面
;
(2)求二面角
的余弦值.
![](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/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7920d2550a6af7df3db60a33fe02c53b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58097af4081e62c2ec10c006828fa544.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/13/74f98797-46d4-4dce-92df-f56008696c3c.png?resizew=163)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcf19a7f0dd0cdf59516ae585025110.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a40e279fbb77437a71f5b5fde83327.png)
您最近一年使用:0次
2023-08-12更新
|
1169次组卷
|
7卷引用:北京市密云区2023届高三考前保温练习(三模)数学试题
名校
解题方法
8 . 如图,在四棱锥
中,底面
为正方形.且
平面
,
,
分别为
,
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/20/b65d4d3e-ec93-4ba2-a533-8d185db999e6.png?resizew=195)
(1)求证:
平面
;
(2)求证:平面
平面
;
(3)若
,求
与平面
所成角的正弦值.
![](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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/20/b65d4d3e-ec93-4ba2-a533-8d185db999e6.png?resizew=195)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83a04565a8ebaa111894b724b0ba266.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/167d31eb8432b5c0364316e5048c23dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
您最近一年使用:0次
名校
9 . 如图,在三棱柱
中,
平面
,
是等腰直角三角形,
,
,
,
分别是棱
,
,
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/19/d3590661-b39b-4150-9dd5-bbdf13c7e4f4.png?resizew=144)
(1)证明:
平面
;
(2)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/002c709e9fee8d477bddfe595cc760f7.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/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/19/d3590661-b39b-4150-9dd5-bbdf13c7e4f4.png?resizew=144)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/810ee7bc82b6f452afb3fc18691abc3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bff3ccea5989c60e51e321af3f53f54.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bff3ccea5989c60e51e321af3f53f54.png)
您最近一年使用:0次
2023-11-22更新
|
255次组卷
|
2卷引用:北京市景山学校2023-2024学年高三上学期期中数学试卷
名校
解题方法
10 . 如图所示,将边长为2的正方形
沿对角线
折起,得到三棱锥
,
为
的中点.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbd1316b9d1f0c1e71fd078deec61f6.png)
(2)再从条件①、条件②这两个条件中选择一个作为已知,求二面角
的余弦值及点
到平面
的距离.
①
;②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b027ed57b9c6f24e27ec0ae282c76efa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccbd1316b9d1f0c1e71fd078deec61f6.png)
(2)再从条件①、条件②这两个条件中选择一个作为已知,求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d7d6e5be7914a224e94a7b7e409a79c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb5255e2159617505e0c87d01437a57.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8825d400f453c5c17a7beeb1cc9a9cf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f2a2c03829634cec6d4d431159c6f27.png)
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