解题方法
1 . 如图所示,在矩形
中,
,
,
为
的中点,沿
将△
翻折,使二面角
为直二面角.
![](https://img.xkw.com/dksih/QBM/2021/12/24/2879458204426240/2880095439290368/STEM/7748ca21-eb0b-481c-a3cb-d5f4de119ac5.png?resizew=431)
(1)求证:
;
(2)求
与平面
所成角的大小;
(3)求二面角
的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb3f9a5da641be35117fd35ba07a6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdd9f345915ae742ed3dcd3f9678264.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f79d7939c88e9702962e5917cad290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97a9b32570d553161be04d13954e92a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f18a490a22cac27417ddc794f00a1941.png)
![](https://img.xkw.com/dksih/QBM/2021/12/24/2879458204426240/2880095439290368/STEM/7748ca21-eb0b-481c-a3cb-d5f4de119ac5.png?resizew=431)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4151e948feebdf7b91fbe739feafa9bc.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6c6e7c025362c46a64a8956761f08e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/164e1cc74e41a2a55d3767c006392bfd.png)
(3)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3db1d8f228c87b65a3609f825fc441d5.png)
您最近一年使用:0次
2021-12-25更新
|
700次组卷
|
2卷引用:广东省广州思源学校2022-2023学年高二上学期10月月考数学试题
名校
解题方法
2 . 如图,四棱锥
的底面是边长为a正方形,每条侧棱长都是底面边长的
倍,P为侧棱SD上的点.
![](https://img.xkw.com/dksih/QBM/2022/5/15/2979504463380480/2981509862113280/STEM/ae5fee007ccb4d81881e29449c9bae47.png?resizew=191)
(1)求证:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c177e06cc3f703e8ca7be7c491fa2942.png)
(2)若
,侧棱SC上是否存在一点E,使得
,若存在,求
的值,若不存在,试说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://img.xkw.com/dksih/QBM/2022/5/15/2979504463380480/2981509862113280/STEM/ae5fee007ccb4d81881e29449c9bae47.png?resizew=191)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c177e06cc3f703e8ca7be7c491fa2942.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb446da7614cc62fd0f5c041ae403807.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e58abb2fa685b7dfe6ca8fbb3543ab6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9013fec36586592a57b9abfb6ce4ffa.png)
您最近一年使用:0次
2022-05-17更新
|
620次组卷
|
6卷引用:广东省佛山市顺德区华侨中学2021-2022学年高二上学期期中数学试题
3 . 如图所示,在平行四边形
中,
,
,
,将△
沿
折起到△
的位置,使平面
平面
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/19/3229f0c6-84bd-488c-8ebb-61de02aaa90f.png?resizew=149)
(Ⅰ)求证:
;
(Ⅱ)若点
为
的中点,求直线
与平面
所成角的正弦值.
![](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/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc11331a7b2d2619b40ee6d34c3bd620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9dfaad4c4467e27421876d8f2a4371d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65277734669566578cbb7d690bb200fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e54cf75bbfc9db93d27937c8b8e977b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/19/3229f0c6-84bd-488c-8ebb-61de02aaa90f.png?resizew=149)
(Ⅰ)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9874eca4abea481fa84eb772a920f9c7.png)
(Ⅱ)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
您最近一年使用:0次
2021-09-11更新
|
334次组卷
|
3卷引用:广东省高州市第七中学等三校2022-2023学年高二上学期11月月考数学试题
4 . 如图,已知四边形
为等腰梯形,
,
,
,P为平面
外一动点,且
为正三角形,G为
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/6098ae63-f14d-4d7f-a90b-8936d971c9c6.png?resizew=177)
(1)证明:
;
(2)若
,当四棱锥
的体积取得最大值时,求平面
与平面
所成的锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8eab423cbac44381618afeacd764d8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6f644b5e0596b96f70fb22705368f20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dceb5cc71fc50f20649f6b9535fd914.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a675310c8ba418e5a59beb7317e21e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/6098ae63-f14d-4d7f-a90b-8936d971c9c6.png?resizew=177)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e06125981292e625d04c6c8dbf89770f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b96fac11d72f72c805dbddb8da72d68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
您最近一年使用:0次
2021-12-28更新
|
900次组卷
|
2卷引用:广东省2022届高三上学期一轮复习联考(四)数学试题
名校
5 . 如下图(左),平面四边形
中,
是
的垂直平分线,垂足为E,
中点为
,
,
,
,沿
将
折起,使C至
位置,如下图(右)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/13/2d1c6cf1-4f81-4292-99a5-332bee83161a.png?resizew=324)
(1)求证:
;
(2)当点
在平面
上的投影为点E时,求直线
与面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70dc2c20619a4fc12a0cfda59af5b69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714cc3707bba3bfdb56e251999be8592.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e320a45fdb0f3339eff27dd0c608117.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661ff55b5ebbadfb600989af3cfce2fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4c8a9c4957431681ddfc77895a88508.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/13/2d1c6cf1-4f81-4292-99a5-332bee83161a.png?resizew=324)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/555dfe77eeb168a880694e22bd9acbdd.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e210795a966bd5dc1293479ab1988b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/de6d1552acf5dbeb80b49f5a1814c0c7.png)
您最近一年使用:0次
名校
6 . 如图,已知三棱柱
,平面
平面
,
,
分别是
的中点.
![](https://img.xkw.com/dksih/QBM/2021/4/14/2699780907016192/2806720330907648/STEM/d81e3ded-334e-4162-8e92-a22d0d5e56ee.png?resizew=346)
(1)证明:
;
(2)求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0671b4776e142e17a79af5b3f0378ef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a059ac387c0ba5972314f00063baa24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba5db30bb3d52a2781a8159ab1c76deb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f82bc0ac36aa52b4d24c35789cc72672.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba75ab97c8b4bb1f6cd7613d7532a5c3.png)
![](https://img.xkw.com/dksih/QBM/2021/4/14/2699780907016192/2806720330907648/STEM/d81e3ded-334e-4162-8e92-a22d0d5e56ee.png?resizew=346)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58cc90fee532e50d319081d571410421.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9afac7c616bbb14e1ed428a3c507c7dc.png)
您最近一年使用:0次
2021-09-12更新
|
1185次组卷
|
8卷引用:广东省广州市执信中学2021届高三上学期第五次月考数学试题
解题方法
7 . 如图,在四棱锥
中,底面四边形
是正方形,且顶点
到
,
,
,
的距离相等,
与
交于点
,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/b81433b8-3423-46f9-b7fa-7aaef185a893.jpg?resizew=183)
(1)求证:
;
(2)若
,求平面
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/b81433b8-3423-46f9-b7fa-7aaef185a893.jpg?resizew=183)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5174d766a4c4105686b9716372c92c3c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a93767331e9bac06a564973a9f4fc663.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9c9cfa597b444b5c9dbae7a825a695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ef796b46e68fe77b117ff0483d2370c.png)
您最近一年使用:0次
2021高三上·广东·专题练习
名校
8 . 如图,在四棱柱
中,底面
是为菱形,
,
平面
,E为
的中点.
![](https://img.xkw.com/dksih/QBM/2021/10/10/2826415523364864/2827586378833920/STEM/b41a6a5890d74ce7b18db7b59a67c392.png?resizew=219)
(1)证明:
;
(2)若
与平面
所成角为
,且
,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/e075468e7fb0bf30229aec01a7205977.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/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://img.xkw.com/dksih/QBM/2021/10/10/2826415523364864/2827586378833920/STEM/b41a6a5890d74ce7b18db7b59a67c392.png?resizew=219)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb5d56d8170b764b80a672cd6c861921.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aec393eec45c4b0d474902b23c3d70bc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5102c216393e133fa25dba98cd78535.png)
您最近一年使用:0次
2021-10-12更新
|
269次组卷
|
4卷引用:数学-学科网2021年高三1月大联考(广东卷)
名校
9 . 如图所示,
是等边三角形,
,
,二面角
为直二面角,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/19/380e3a43-8c39-4b5d-882a-ccff3ee9a44b.png?resizew=148)
(1)求证:
;
(2)求平面
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1912c4e0fe478ba402a887fdc759d3f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13a8d8fe0503af1c8f8d04eaf211bac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f0ac3005d5ecd6d4cea0ce99a47ef3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53407e6605074b8071d2ce57900a71f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/19/380e3a43-8c39-4b5d-882a-ccff3ee9a44b.png?resizew=148)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58cc90fee532e50d319081d571410421.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b9d54cbbf601f4583659771eb534997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87c0bfeadcf17b2a45896071f07a4a5a.png)
您最近一年使用:0次
名校
10 . 如图1,在平行四边形
中,
=60°,
,
,
,
分别为
,
的中点,现把平行四边形
沿
折起如图2所示,连接
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/3/b8e39710-99eb-4ef5-abe1-9a672673aa4c.png?resizew=396)
(1)求证:
;
(2)若
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/883732ae71bfed76e07732ec709f4653.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b5d693c4f0c4d0e6c0c810e7d464b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92535536bd3c2761724fd058427f95a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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2021-06-15更新
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12卷引用:广东省广州市广州大学附属中学2021-2022学年高二上学期第一次月考数学试题
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