名校
1 . 如图,在三棱锥
中,侧面
底面
,
,
是边长为2的正三角形,
,
分别是
的中点,记平面
与平面
的交线为
.
平面
;
(2)设点
在直线
上,直线
与平面
所成的角为
,异面直线
与
所成的角为
,求当
为何值时,
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615fc8790237a1b09af51d6bcad6b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fbfcae2cecc98e2d6c16dde6d3ec1c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/829a1a887ceba13dd8551b1e3604bf6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9df740160690029ac1e730c85f20347.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
(2)设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4820decdfaf6808eda1b625cc8aa0110.png)
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6卷引用:重庆市第八中学2022届高三下学期调研检测(七)数学试题
重庆市第八中学2022届高三下学期调研检测(七)数学试题湖南师范大学附属中学2022届高三下学期月考(七)数学试题山西大学附属中学校2023届高三下学期3月模块诊断数学试题云南省红河州建水实验中学2022-2023学年高一下学期4月考试数学试题(已下线)专题03 空间向量求角度与距离10种题型归类-【巅峰课堂】2023-2024学年高二数学上学期期中期末复习讲练测(人教A版2019选择性必修第一册)湖北省宜荆荆2024届高三下学期五月高考适应性考试数学试题
名校
解题方法
2 . 如图,在多面体
中,平面
与平面
均为矩形且相互平行,
,设
.
平面
;
(2)若多面体
的体积为
:
(i)求
;
(ii)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17d4a6cf11cda87b3dfafaecdec683f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dff64de03b0302dbc12f2fc207b70d1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/336e0a8f5fbc1c44a02adab5a1fffb60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dc99203b785fbdbd399bb03c7556fbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
(2)若多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17d4a6cf11cda87b3dfafaecdec683f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a391005600bdd69c96750589f9adb048.png)
(i)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
(ii)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffee8b7eff437080a0936d837ceabe95.png)
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2卷引用:重庆市乌江新高考协作体2023-2024学年高二下学期第二阶段性学业质量联合调研抽测(5月)数学试题
名校
解题方法
3 . 如图,在四棱锥
中,底面
为直角梯形,
,
平面
,
,点
分别在线段
和
的中点.
平面
;
(2)求平面
与平面
夹角.
![](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/405effb49ef901476701e72cc47918da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f95475bfc06e884754eb4a455c3f434e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd706c0e3aa382425502a1262dc6b735.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c7c261740ac2ae26715e1298ca278a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab5f236e0c248607721ff77b6ea8b6ee.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab5f236e0c248607721ff77b6ea8b6ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b81fb655624ff75a5eab94de9b8c8e9.png)
您最近一年使用:0次
4 . 如图,在四棱锥
中,底面
是梯形,
,侧面
为正三角形,且与底面
垂直,E为
的中点,M在
上,满足
.
时,证明:
平面
;
(2)当二面角
为
时,求
的值.
![](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/642afba05807cdacf3ed4057d5dec3b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](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/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62634b76241cef6e250eee87dad489c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73b3cf0f585938ede9eca890a6eb326d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f0b39562ebcbac4476e41725a66bb4c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
(2)当二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e8eea7f5526dba4c58723366b36c6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15615de1a6df206dbd081251f676578e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
您最近一年使用:0次
名校
解题方法
5 . 如图,四棱锥
中,底面ABCD为正方形,
面ABCD,
,E,F分别是PC,AD的中点.
平面PFB;
(2)求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99926bf272cd757f0985c69b390ebcce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/063510e3c1fb6a7ccc3b8e3e3c7d660e.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c3425aee6c70e3c522b95e2a4e2b07.png)
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4卷引用:重庆市七校联盟2023-2024学年高一下学期5月期中联合考试数学试题
重庆市七校联盟2023-2024学年高一下学期5月期中联合考试数学试题2015-2016学年江西省赣州市高二上学期期末文科数学试卷2016-2017学年江西丰城中学高二上月考一数学(文)试卷(已下线)专题05 立体几何初步(2)-期末考点大串讲(苏教版(2019))
名校
解题方法
6 . 如图所示,在长方形
中,
,
为
的中点,以
为折痕,把
折起到
的位置,且平面
平面
.
;
(2)求四棱锥
的体积;
(3)在棱
上是否存在一点P,使得![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac8b40d14544a9be0bebdb276f0fa865.png)
平面
,若存在,求出点P的位置;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c79e56bc6f1db8f446fc5bd34a08865.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63b43490ca09467a4c8cd8cfe91c94e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84f9c64303370347131dd9d8c5c70c01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70fd4f68511d2393905617bfdeddddec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01ff27eea7545bb06f9472f91290c54e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c75d1b97dc32e2b99bccd4d8a02ef17.png)
(2)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f7d528d7d5aea71bb3d9df16055c2a7.png)
(3)在棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d226aef207ce71a381d6f63801cc9d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac8b40d14544a9be0bebdb276f0fa865.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d67be899bc131ec1b9921ae9787c40d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
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10卷引用:重庆市永川北山中学校2022-2023学年高一下学期期中数学试题
重庆市永川北山中学校2022-2023学年高一下学期期中数学试题安徽省合肥市第十一中学2020-2021学年高二上学期第一次月考数学(理)试题(已下线)【新东方】杭州新东方高中数学试卷324江西省抚州市金溪县第一中学2020-2021学年高二上学期入学考试数学(理)试题湖南省邵阳市邵东市第三中学2020-2021学年高一下学期第三次月考数学试题广东省东莞市东莞实验中学2022-2023学年高一下学期5月月考数学试题广东省东莞市东莞高级中学2022-2023学年高一下学期期中数学试题(已下线)第八章 本章综合--考点强化训练【第一练】“上好三节课,做好三套题“高中数学素养晋级之路(已下线)专题06 立体几何初步解答题热点题型-《期末真题分类汇编》(江苏专用)(已下线)专题04 第八章 立体几何初步(2)-期末考点大串讲(人教A版2019必修第二册)
名校
7 . 如图所示的几何体是一个半圆柱和一个三棱锥的组合体.
是半圆柱的母线,
分别是底面直径BC和
的中点,
是半圆
上一动点,
是半圆
上的动点,
是圆柱的母线,延长
至
点使得
为
的中点,连接
,
构成三棱锥
.
;
(2)当三棱锥
的体积最大时,求平面
与平面
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ebf67d880172b27fefacc3c5b808eae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4d6cde991efd85f03f8801e745127a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683e17492ad37eeab880209057cb0feb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a696a182fff038a86b2bbe8ca099442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30513ea48bc1ef3ae78adac83d894f14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4d260c4df7b0dc180af6980d21f3371.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf5ae8d145c5ce43e4cfc95fe6f563ed.png)
(2)当三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e367b683581c7cbe018078168f69efc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b36a70bc52a720ba8750aee4924307.png)
您最近一年使用:0次
名校
8 . 如图,在四棱锥
中,底面
是平行四边形,
分别为
的中点,
为线段
上一点,且
.
平面
;
(2)若四棱锥
为正四棱锥,且
,求四棱锥
的外接球与正四棱锥
的体积之比.
![](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/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e7344dca1e40bf072371ddd5640111.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e96d954fad9d528c69a21129837431cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b8badfeb9e7556486e02ab60df4dd32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4734735213b599a9915e1ed91a5d8ce4.png)
(2)若四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9d5d99f272872783fce8189096298d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
您最近一年使用:0次
名校
9 . 如图,
为圆锥的顶点,
是圆锥底面的圆心,AC为底面直径,
为底面圆
的内接正三角形,且边长为
,点
在母线PC上,且
.
平面BDE;
(2)求二面角
平面角的正弦值;
(3)若点M为线段PO上的动点,当直线
平面ABE时,求AM与平面ABE所成的角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a81dda4896c42a956051e6999ca0f8aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a929faf549d8b8f1cd36d7a98257ace.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/816b7f285cc55bbe5bf873538ba87230.png)
(3)若点M为线段PO上的动点,当直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d0492b25f10ae45c39f8e9838519259.png)
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