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题型:
难度:
分类:
2023高二上·上海·专题练习
解题方法
1 . 如图所示的几何体中,四边形为正方形,
.
![](/uploads/image/idq21133stem/f4a30d0e07484bd7a7a93033f128b70c.png)
(1)求证:![](/uploads/image/squformula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
平面
;
(2)若
,平面
平面.若
为
中点,求证:
.
![](/uploads/image/squformula/6ab1006748f0a9c2181e1144f9a7d9c1.png)
![](/uploads/image/idq21133stem/f4a30d0e07484bd7a7a93033f128b70c.png)
(1)求证:
![](/uploads/image/squformula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](/uploads/image/squformula/fb31ef428bd9de9bc875b343feded3c7.png)
![](/uploads/image/squformula/10fc7991ea17d54ff5f4445ac5699463.png)
(2)若
![](/uploads/image/squformula/a775a5f8a5f08c08c67a1e5eaf8c823c.png)
![](/uploads/image/squformula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](/uploads/image/squformula/a0ed1ec316bc54c37c4286c208f55667.png)
![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](/uploads/image/squformula/f392902d611863c6908a48e696e7bd8f.png)
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名校
解题方法
2 . 如图,已知在四棱锥
中,
平面,点q在棱
上,且
,底面为直角梯形,
,
,
,
,m,n分别是,
的中点.
![](/uploads/image/idqe2127/1137929a-d350-4653-a73a-9c6b41b913bd.png)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值.
![](/uploads/image/squformula/0585b6c0f156eecf9662b9846d4eb693.png)
![](/uploads/image/squformula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](/uploads/image/squformula/bd33764ff4efddfe11a98a609753715c.png)
![](/uploads/image/squformula/1804cb8fd0704cc8eaed0304a9915eb1.png)
![](/uploads/image/squformula/b2651ebf7e1d8f609b4e1aff4b39e2d2.png)
![](/uploads/image/squformula/fcd0ced286a0fbc7e4862f8147264277.png)
![](/uploads/image/squformula/037b342a682cbd4241855a243da3c016.png)
![](/uploads/image/squformula/a88c44f558705de3bcefcfc0ece96b8f.png)
![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](/uploads/image/idqe2127/1137929a-d350-4653-a73a-9c6b41b913bd.png)
(1)求证:
![](/uploads/image/squformula/f8014e499e7852b587b3b36af14b7816.png)
![](/uploads/image/squformula/0b60870baa5e3fbc33a749aa5f0a94be.png)
(2)求直线
![](/uploads/image/squformula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](/uploads/image/squformula/101d5eb54d3f629a378bfd5324f554dd.png)
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1卷引用:云南省昆明市云南师范大学附属中学2023-2024学年高二上学期期末模拟测试数学试题
解题方法
3 . 如图,在边长为4的正三角形
中,
、
分别为边
、
的中点,将
沿
翻折至
,得四棱锥
,设
为
的中点.
![](/uploads/image/idqe2125/0ab45639-1f53-421d-86b6-52509fd4756f.png)
(1)证明:
平面
;
(2)若平面
平面
,求平面
与平面
夹角的余弦值.
![](/uploads/image/squformula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](/uploads/image/squformula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](/uploads/image/squformula/a0ed1ec316bc54c37c4286c208f55667.png)
![](/uploads/image/squformula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](/uploads/image/squformula/60ef95894ceebaf236170e8832dcf7e3.png)
![](/uploads/image/squformula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](/uploads/image/squformula/49b50357a6545cae8348e3059312f520.png)
![](/uploads/image/squformula/eb2d555062f34d5a74f6d47da4ea8888.png)
![](/uploads/image/squformula/5318c615807c26bb0b014d1d4bea6144.png)
![](/uploads/image/squformula/dad2a36927223bd70f426ba06aea4b45.png)
![](/uploads/image/squformula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](/uploads/image/idqe2125/0ab45639-1f53-421d-86b6-52509fd4756f.png)
(1)证明:
![](/uploads/image/squformula/eb3a99c3ef79e88a8ecbd6824fc7ae7f.png)
![](/uploads/image/squformula/923189afc198d153c79059a827f63c87.png)
(2)若平面
![](/uploads/image/squformula/004dd8ad9e5a200b3869ebfc59c2446d.png)
![](/uploads/image/squformula/a03952d664fba91020fc5f3bcf2f9746.png)
![](/uploads/image/squformula/5395f1811518a917a30e5949c4c8fc57.png)
![](/uploads/image/squformula/a09d9d486b7f91ba933210dd013a7f2c.png)
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1卷引用:广东省汕头市2024届高三上学期期末调研测试数学试题
解题方法
4 . 如图,在四棱锥
中,底面为菱形,
,
底面,
,
,
,
分别是
,
,
的中点.
![](/uploads/image/idqe2126/372ab7fc-bf36-4f67-a672-4eb05330b1bb.png)
(1)求证:
平面
;
(2)求点
到平面
的距离.
![](/uploads/image/squformula/0585b6c0f156eecf9662b9846d4eb693.png)
![](/uploads/image/squformula/c6e0b64d25ddd18454f88e40c45d7d8f.png)
![](/uploads/image/squformula/5a1b49f64e0065edad868b25e9fcada3.png)
![](/uploads/image/squformula/40d4d36ae30487030b827ce9413b9f13.png)
![](/uploads/image/squformula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](/uploads/image/squformula/a0ed1ec316bc54c37c4286c208f55667.png)
![](/uploads/image/squformula/895dc3dc3a6606ff487a4c4863e18509.png)
![](/uploads/image/squformula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](/uploads/image/squformula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](/uploads/image/idqe2126/372ab7fc-bf36-4f67-a672-4eb05330b1bb.png)
(1)求证:
![](/uploads/image/squformula/7c27a8fd3bf5b89a16dbbe1a8230653c.png)
![](/uploads/image/squformula/852aabd89edffc1b94344ff3f1f31ccd.png)
(2)求点
![](/uploads/image/squformula/895dc3dc3a6606ff487a4c4863e18509.png)
![](/uploads/image/squformula/1e582d73b96ba649378379c3074d506d.png)
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1卷引用:内蒙古自治区赤峰市2020-2021学年高一下学期期末联考文科数学试题(a)
智能选题,一键自动生成优质试卷~
5 . 如图,四棱锥
中,底面为平行四边形,
,
,
底面.设
中点为
,
中点为
.
![](/uploads/image/idqe2125/98fa0bec-acb2-4fbd-848c-52b65685de63.png)
(1)求证:
平面
;
(2)若
,求直线
与面
所成的角的正弦值.
![](/uploads/image/squformula/0585b6c0f156eecf9662b9846d4eb693.png)
![](/uploads/image/squformula/c6e0b64d25ddd18454f88e40c45d7d8f.png)
![](/uploads/image/squformula/4c4e4a162f12d12a082b8d8fdd1aeab9.png)
![](/uploads/image/squformula/5a1b49f64e0065edad868b25e9fcada3.png)
![](/uploads/image/squformula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](/uploads/image/squformula/ac047e91852b91af639feec23a9598b2.png)
![](/uploads/image/squformula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](/uploads/image/squformula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](/uploads/image/idqe2125/98fa0bec-acb2-4fbd-848c-52b65685de63.png)
(1)求证:
![](/uploads/image/squformula/edcf19a7f0dd0cdf59516ae585025110.png)
![](/uploads/image/squformula/852aabd89edffc1b94344ff3f1f31ccd.png)
(2)若
![](/uploads/image/squformula/ba4c15fb8fc3239d45bd4e7d8971f58e.png)
![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](/uploads/image/squformula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
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1卷引用:内蒙古自治区赤峰市2020-2021学年高一下学期期末联考理科数学试题(a)
解题方法
6 . 已知正方体
的棱长为2,
为
的中点,
为
的中点.
![](/uploads/image/idqe2125/2d3c2fe7-191d-4dc2-8387-2bcf90b43e24.png)
(1)求证:
平面
;
(2)求平面
与平面
夹角的余弦值.
![](/uploads/image/squformula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](/uploads/image/squformula/ac047e91852b91af639feec23a9598b2.png)
![](/uploads/image/squformula/6655cc150ddc9deba2254780984d0024.png)
![](/uploads/image/squformula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](/uploads/image/squformula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](/uploads/image/idqe2125/2d3c2fe7-191d-4dc2-8387-2bcf90b43e24.png)
(1)求证:
![](/uploads/image/squformula/f214481e6b23307a37940f6dd0313d30.png)
![](/uploads/image/squformula/5592038271a0fcc886d12fd953de4e6b.png)
(2)求平面
![](/uploads/image/squformula/5592038271a0fcc886d12fd953de4e6b.png)
![](/uploads/image/squformula/0d609847e2ff3d64e5a514582c3ead0e.png)
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1卷引用:内蒙古呼和浩特市2024届高三上学期学业质量监测数学(理)试题
解题方法
7 . 如图,在四棱锥
中,已知底面为矩形,
平面
为棱的中点,连接
.求证:
![](/uploads/image/idqe2125/a31b7a1b-f815-4eef-9343-61722d992603.png)
(1)![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
平面
;
(2)平面
平面.
![](/uploads/image/squformula/0585b6c0f156eecf9662b9846d4eb693.png)
![](/uploads/image/squformula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](/uploads/image/squformula/1417f51d619f9b9f86154fdbf12e55e1.png)
![](/uploads/image/squformula/c367e865cab70191ca538976d0fedde3.png)
![](/uploads/image/idqe2125/a31b7a1b-f815-4eef-9343-61722d992603.png)
(1)
![](/uploads/image/squformula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](/uploads/image/squformula/895d6f710d5f67e1d4c7408d50d77281.png)
![](/uploads/image/squformula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
(2)平面
![](/uploads/image/squformula/93edc7bb513f40a89173121c8570cd65.png)
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1卷引用:内蒙古赤峰市2021届高三上学期12月双百金科大联考数学(文)试题
8 . 如图,在四棱锥
中,
为等边三角形,
,
,
,e,f分别是bc,pd的中点.
![](/uploads/image/idqe2125/ca3cb9b1-39af-4a43-ad89-9fd9c5d0cce5.png)
(1)证明:
平面pab.
(2)若
,求平面aef与平面pbd夹角的余弦值.
![](/uploads/image/squformula/0585b6c0f156eecf9662b9846d4eb693.png)
![](/uploads/image/squformula/05e78042a384255038de485fd7bc0839.png)
![](/uploads/image/squformula/5f79863ffcfa63117ca6741b20a48e69.png)
![](/uploads/image/squformula/1134c8e3440abb6cd385af2c169037fe.png)
![](/uploads/image/squformula/bc692e220c54f56f00bcd67b4499d5db.png)
![](/uploads/image/idqe2125/ca3cb9b1-39af-4a43-ad89-9fd9c5d0cce5.png)
(1)证明:
![](/uploads/image/squformula/06222ee533c2484ab25321a6abbf98cb.png)
(2)若
![](/uploads/image/squformula/6bbd7c2767c106faf27d6a97ebc8e739.png)
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2卷引用:辽宁省辽阳市2024届高三上学期期末数学试题
名校
解题方法
9 . 如图,矩形中
为边
的中点,将
沿直线
翻折成
,使
,若
为线段
的中点,
![](/uploads/image/idqe2124/599a3b3e-4d98-4b40-a343-500f0080a1f0.png)
(1)求证:
平面![](/uploads/image/squformula/657dffbd3623b705f871878fbd9df57e.png)
(2)求证:平面
平面![](/uploads/image/squformula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
(3)求二面角
夹角的正弦值
![](/uploads/image/squformula/9d7a909494a000db960624410c5eee24.png)
![](/uploads/image/squformula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](/uploads/image/squformula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](/uploads/image/squformula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](/uploads/image/squformula/3aa2a83fed9bf4cb09d84a980452e346.png)
![](/uploads/image/squformula/642e15f06c38176c827db801333153d2.png)
![](/uploads/image/squformula/ac047e91852b91af639feec23a9598b2.png)
![](/uploads/image/squformula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](/uploads/image/idqe2124/599a3b3e-4d98-4b40-a343-500f0080a1f0.png)
(1)求证:
![](/uploads/image/squformula/5f369bec2d5682bf6b8b317a08aff546.png)
![](/uploads/image/squformula/657dffbd3623b705f871878fbd9df57e.png)
(2)求证:平面
![](/uploads/image/squformula/d7b3e7c7845a0ec3cbac709fda131764.png)
![](/uploads/image/squformula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
(3)求二面角
![](/uploads/image/squformula/db26c5df2acdf210f3426fad185b1c20.png)
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1卷引用:黑龙江省牡丹江市第二子共同体2024届高三上学期期末联考数学试题
名校
解题方法
10 . 如图,在正三棱柱
中,
为
的中点.
![](/uploads/image/idqe2124/effd6aa0-ce25-4872-816b-6025d968c6f8.png)
(1)证明:
平面
;
(2)若
,求直线
与平面
所成角的正弦值.
![](/uploads/image/squformula/42d3a82b8e587ee890467835bc4e854c.png)
![](/uploads/image/squformula/8455657dde27aabe6adb7b188e031c11.png)
![](/uploads/image/squformula/60ef95894ceebaf236170e8832dcf7e3.png)
![](/uploads/image/idqe2124/effd6aa0-ce25-4872-816b-6025d968c6f8.png)
(1)证明:
![](/uploads/image/squformula/1496afecd92a619fbe5e9b736f06f4e3.png)
![](/uploads/image/squformula/a935b7d21a103a264b6e96ecf82dbe4a.png)
(2)若
![](/uploads/image/squformula/eeed487430a5b8a330f2d0c52166521a.png)
![](/uploads/image/squformula/0d8772aa893a9c1d40f714cb25701701.png)
![](/uploads/image/squformula/6ac61c24f99a4e466f1e2ea011893866.png)
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1卷引用:重庆市黔江中学校2021-2022学年高二上学期9月月考试数学试题
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