名校
1 . 如图,在直三棱柱
中,
,
,
为
的中点.
平面
.
(2)若以
为直径的球的表面积为
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95578eba5dd34ca64b5f228640819cd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b531aaca9d037a0d047511eec8f350ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/504a36c231b8e80724d01649e7c0944f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41104641f3e2260d00aeadf8fb8a078a.png)
(2)若以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b470c4e195cf7a07b7a331ce4b436e03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95265f94a8eb7f76b5db6875246a091d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee527f97d0bfc89f791b728d80e562d3.png)
您最近一年使用:0次
2024-04-20更新
|
1392次组卷
|
3卷引用:广东省湛江市2024届高三下学期二模考试数学试题
2 . 如图,在三棱锥
中,平面
平面
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/26/4334befb-686d-4f51-9731-2c955e5fb317.png?resizew=177)
(1)求三棱锥
外接球的表面积;
(2)设D为侧棱
上一点,若二面角
的大小为
,证明:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a3fd5284e160896f07ce367645fd04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33b10cffb8dc140dd2cb4202cd537dd8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/26/4334befb-686d-4f51-9731-2c955e5fb317.png?resizew=177)
(1)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
(2)设D为侧棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f1854ba6cc92481d7a616bd2788a47e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb5d56d8170b764b80a672cd6c861921.png)
您最近一年使用:0次
2023·全国·模拟预测
解题方法
3 . 如图,球O是正三棱锥
和
的外接球,M为
的外心,直线AM与线段BC交于点D,D为BC的中点,两三棱锥的高之比为
,E为PA上一点,且
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/23/6b6f5eb2-db3b-46ee-b7d2-562daea12a75.png?resizew=163)
(1)证明:
;
(2)求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7edc2e23df190c35aafad93410a05b8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3db26d5fde5cc9ec365d155a13d8c725.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a17ed1752f3221f1e9a10323c555a6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/23/6b6f5eb2-db3b-46ee-b7d2-562daea12a75.png?resizew=163)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f22f74a1995a5b5fa2a0536606ce1df.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b2492f61e8b8a5699f2ad48ffb93082.png)
您最近一年使用:0次
名校
4 . 如图,在四棱锥
中,平面
平面
,
,
,
,
.
![](https://img.xkw.com/dksih/QBM/2022/5/3/2971556730806272/2973106767257600/STEM/79650ada-784b-4eee-9e0a-188f9385105b.png?resizew=203)
(1)求证:
平面
;
(2)若三棱锥
的外接球表面积为
,求三棱锥
的体积与三棱锥
的外接球的体积的比值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78a3fd5284e160896f07ce367645fd04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00b9a5e110c068c60dd41f95bce4ab1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f571396be1aa4a8914a66f7d7abd6381.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af260e0d98c95d1e092dc4c6d348e3ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34c9dda5e21318e671d6f1cc10f2ed32.png)
![](https://img.xkw.com/dksih/QBM/2022/5/3/2971556730806272/2973106767257600/STEM/79650ada-784b-4eee-9e0a-188f9385105b.png?resizew=203)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97f30533da2e1d2a958dc906c37eba9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f571a1aac46c6d0cf440c0ec2846bf9.png)
(2)若三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b172e3aae625013716b30fae2c59279.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95345846d2dd4dfa042a9093c62a8b82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b183492677d0457b8701c53d9fa1414.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b183492677d0457b8701c53d9fa1414.png)
您最近一年使用:0次
名校
5 . 如图所示,直三棱柱
的所有棱长均相等,点
为
的中点,点
为
的中点.
![](https://img.xkw.com/dksih/QBM/2022/4/27/2967030284017664/2967848002502656/STEM/a6999637-3ab4-41bd-b4a7-335865dba36e.png?resizew=148)
(1)求证:
平面
;
(2)若三棱锥
的体积为
,求该三棱柱的外接球表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7f6f93171329d508d491143b9d71f7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f1f229274a6e17977cc047814212589.png)
![](https://img.xkw.com/dksih/QBM/2022/4/27/2967030284017664/2967848002502656/STEM/a6999637-3ab4-41bd-b4a7-335865dba36e.png?resizew=148)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75aea24647cd4d0b4b9aa513bf5457b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9edc50f7febbc2d5d8dcdc23a3630a7.png)
(2)若三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b42e4d3ce220fd60e952c957fb71a6d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18483c9c195ecd922772527fa85c0fcb.png)
您最近一年使用:0次
2022-04-28更新
|
819次组卷
|
3卷引用:安徽省六安市舒城中学2022届高三下学期仿真模拟(三)文科数学试题
安徽省六安市舒城中学2022届高三下学期仿真模拟(三)文科数学试题山西省大同市第三中学校2021-2022学年高一下学期期中数学试题(已下线)第二章 立体几何中的计算 专题六 几何体的外接球、棱切球、内切球 微点4 圆柱、直三棱柱及其切割体模型综合训练【基础版】
名校
6 . 如图,在梯形ABCD中,
,
,
,E为AD的中点,以BE为折痕将
折起,使点A到达点P的位置,连接PD,PC.
![](https://img.xkw.com/dksih/QBM/2022/4/30/2969167802540032/2973770861920256/STEM/d3208572-15fe-43c7-879b-c1edee8aeca6.png?resizew=264)
(1)证明:平面
平面BCDE;
(2)当
时,若几何体
的顶点均在球O的表面上,求球O的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d730ae4307db56b47849c3a19dedfb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88f86b6bb8d0612e06f5579090727379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e13c772461aef1d9d715129636739748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://img.xkw.com/dksih/QBM/2022/4/30/2969167802540032/2973770861920256/STEM/d3208572-15fe-43c7-879b-c1edee8aeca6.png?resizew=264)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a940f43e94a687339a9b50e0694e2e8f.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2899e607479d8d1c47d954ae9ebb7144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e98920101c174b991d7a8481707ab88.png)
您最近一年使用:0次
7 . 如图,已知正三棱锥
中,
,
,VD⊥平面ABC,垂足为D,DE⊥平面VAB,垂足为E,连接VE并延长,交AB于点M.
![](https://img.xkw.com/dksih/QBM/2022/3/4/2929119787147264/2932689291313152/STEM/03babc6a-06e4-4f6d-a59c-98076a7c2681.png?resizew=221)
(1)证明:M是AB的中点;
(2)过点E作EF⊥平面VAC,垂足为F,求四面体VDEF的外接球的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6a94d59dee2d5a8f0425b64b2083825.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f59502f452fb6a290484608e65a412df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18f3952d6946970a965cdefa3e75797c.png)
![](https://img.xkw.com/dksih/QBM/2022/3/4/2929119787147264/2932689291313152/STEM/03babc6a-06e4-4f6d-a59c-98076a7c2681.png?resizew=221)
(1)证明:M是AB的中点;
(2)过点E作EF⊥平面VAC,垂足为F,求四面体VDEF的外接球的体积.
您最近一年使用:0次
2022-03-09更新
|
291次组卷
|
2卷引用:陕西省延安市宜川县中学2023届高三一模文科数学试题
8 . 如图,在正三棱锥
中,
是高
上一点,
,直线
与底面所成角的正切值为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/19/e1b3bcd4-8d48-4a1f-8af8-21c2829c73fb.png?resizew=140)
(1)求证:
平面
;
(2)求三棱锥
外接球的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cabb973891c409b9b43ff339978f618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/19/e1b3bcd4-8d48-4a1f-8af8-21c2829c73fb.png?resizew=140)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f4c3f9dd5d0343597a7f58a1989b537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8257b6bd25104e07b9ad935c0a3aac4.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51f73a0ca4e6c794242489066fddb6c5.png)
您最近一年使用:0次
2021-06-26更新
|
1058次组卷
|
4卷引用:江苏省南通密卷2021届高三模拟试卷数学试题
江苏省南通密卷2021届高三模拟试卷数学试题安徽省滁州市凤阳县临淮中学2022届高三下学期三模文科数学试题重庆市2023届高三五月第二次联考数学试题(已下线)1.2.2 空间中的平面与空间向量(分层训练)-2023-2024学年高二数学同步精品课堂(人教B版2019选择性必修第一册)
9 . 如图所示,在四棱锥
中,底面
是边长等于2的正方形,且平面
平面
,
,若四棱锥
的高等于1.
![](https://img.xkw.com/dksih/QBM/2021/4/6/2693852784238592/2716166946258944/STEM/70b65407-65f5-403d-acb0-d48e24c35c99.png?resizew=251)
(1)求证:平面
平面
;
(2)求四棱锥
外接球的体积.
![](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/19ad6a0124359e8b9f7649cf0bff51ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53c8a72acdef14452a6c62f2a60a15fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://img.xkw.com/dksih/QBM/2021/4/6/2693852784238592/2716166946258944/STEM/70b65407-65f5-403d-acb0-d48e24c35c99.png?resizew=251)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f3e2bed5ce5fe466395d2f5743d335b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb5255e2159617505e0c87d01437a57.png)
(2)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
您最近一年使用:0次
名校
解题方法
10 . 如图所示,在矩形
中,
,
,点
是线段
的中点,把三角形
沿
折起,设折起后点
的位置为
,
是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/24eb02e2-6f15-4613-b94f-1b2f33386b8a.png?resizew=206)
(1)求证:无论
在什么位置,都有
平面
;
(2)当点
在平面
上的射影落在线段
上时,若三棱锥
的四个顶点都在一个球上,求这个球的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f4aca5534bce25acaeb7379deed8f8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97a9b32570d553161be04d13954e92a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/18/24eb02e2-6f15-4613-b94f-1b2f33386b8a.png?resizew=206)
(1)求证:无论
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dcafa398cc6b6079883e7ad153eb62d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0bf12905647aeeded72bbca21a63f319.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f11bfca0b64b54b4b804e460162dc81.png)
您最近一年使用:0次