名校
1 . 如图,平行六面体
的底面
是矩形,P为棱
上一点.且
,F为
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/5/27bbb2b8-53cd-4f1d-9612-ff0d4ae66513.png?resizew=193)
(1)证明:
;
(2)若
.当直线
与平面
所成的角为
,且二面角
的平面角为锐角时.求三棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63e4d19bf237a6fca67e0d01a9ddb726.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/5/27bbb2b8-53cd-4f1d-9612-ff0d4ae66513.png?resizew=193)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a56375c3423cc022ac9d6d04e3a61bb9.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a1c9d4808c72fb8e4c885e236d62967.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1636b4530c0b42d0e0b649e90e3b9e85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3379b0519d41af970b9d28dfba0094d8.png)
您最近一年使用:0次
2022-04-08更新
|
1249次组卷
|
2卷引用:河北省石家庄市2022届高三二模数学试题
2 . 四面体
中,
(1)
.求证:这个四面体的四个面都是锐角三角形;
(2)有4条长为2的线段和2条长为
的线段,用这6条线段作为棱,构成一个三棱锥,问
为何值时,可构成一个最大体积的三棱锥,最大值为多少?
(参考公式:三元均值不等式
,当且仅当
时取得等号)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f319885045519b00c26344a924506d96.png)
(2)有4条长为2的线段和2条长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
(参考公式:三元均值不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1646a98010bb9876a1d249ff2bd11ccc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44acc0ee22dc4b7750e8be825e7c1355.png)
您最近一年使用:0次
名校
3 . 如图,在四棱锥
中,底面
为直角梯形,
,
,
,
,
为正三角形,点
,
分别在线段
和
上,且
.设二面角
为
,且
.
![](https://img.xkw.com/dksih/QBM/2021/6/24/2750004782448640/2781075118645248/STEM/7c97cefa-f451-4d48-b010-78c27ffe43f0.png?resizew=277)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值;
(3)求三棱锥
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/f571396be1aa4a8914a66f7d7abd6381.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4795ee1f96b430529934e2231b38885d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f807fa55d6a411a31cd1c6bc8cffe59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c025ee3317be1099b7bf03a11e37ed4.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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c4e32e152097c2dfad9769da74680b6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c911b404bbb8f8d5f1470585fa31ad97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e49129bc80bb9b119c94d81deb177f.png)
![](https://img.xkw.com/dksih/QBM/2021/6/24/2750004782448640/2781075118645248/STEM/7c97cefa-f451-4d48-b010-78c27ffe43f0.png?resizew=277)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c19129982fd8389238b303e091bd94c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bfaf581b4f42a25087f7eee23a7d66b6.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(3)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78c5822ecaac92df0e7e2562b5670df5.png)
您最近一年使用:0次
2021-08-07更新
|
1053次组卷
|
2卷引用:江苏省常州市2020-2021学年高一下学期期末数学试题
4 . 棱锥是生活中最常见的空间图形之一,譬如我们熟悉的埃及金字塔,它的形状可视为一个正四棱锥.我国数学家很早就开始研究棱锥问题,公元一世纪左右成书的《九章算术》第五章中的第十二题,计算了正方锥、直方锥(阳马)、直三角锥(鳖臑)的体积,并给出了通用公式.公元三世纪中叶,数学家刘徽在给《九章算术》作的注中,运用极限思想证明了棱锥的体积公式.请你使用学过的相关知识,解决下列问题:如图,正三棱锥
中,三条侧棱SA,SB,SC两两垂直,侧棱长是3,底面
内一点P到侧面
的距离分别为x,y,z.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/11c42ecf-75a2-4e0c-94f1-c24e2f1d0cc1.png?resizew=147)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/307782c7-fe75-4f04-a03a-b35f73bbb313.png?resizew=135)
(1)求证:
;
(2)若
,试确定点P在底面
内的位置.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef3d23ea0b4df6b49a03636e71cb9504.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/11c42ecf-75a2-4e0c-94f1-c24e2f1d0cc1.png?resizew=147)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/307782c7-fe75-4f04-a03a-b35f73bbb313.png?resizew=135)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c2a362b51ffc3fdc60a15a394eeb98.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fcf59e219a162bb6611dd7e7d7b0893.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
您最近一年使用:0次
名校
解题方法
5 . 如图,在直三棱柱ABC-A1B1C1中,∠ABC=90 °,AB=BC=AA1=2,M,N分别是棱BC,A1C1的中点,点P在线段B1N上,
,AC1交A1C于点S,若PS∥面B1AM.
![](https://img.xkw.com/dksih/QBM/2021/4/14/2699759241822208/2699806607523840/STEM/e480c76a-d0b2-442e-8ea5-c5774f79e47e.png?resizew=215)
(1)证明: PS//B1Q;
(2)求三棱锥P- B1AM的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8e805c6e38405c84691637706e8d5e2.png)
![](https://img.xkw.com/dksih/QBM/2021/4/14/2699759241822208/2699806607523840/STEM/e480c76a-d0b2-442e-8ea5-c5774f79e47e.png?resizew=215)
(1)证明: PS//B1Q;
(2)求三棱锥P- B1AM的体积.
您最近一年使用:0次
6 . 如图,已知四棱锥
,
且
,
,
,
,
的面积等于
,E是PD是中点.
![](https://img.xkw.com/dksih/QBM/2021/6/29/2753469786300416/2781031193870336/STEM/37dc3ed6-8a66-441e-993b-dff3af0ce8c3.png?resizew=282)
(Ⅰ)求四棱锥
体积的最大值;
(Ⅱ)若
,
.
(i)求证:
;
(ii)求直线CE与平面PBC所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef699f5dc072b853cfe700c6f1abbbae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a8751f226cdfbff4119a12c75a8df30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d768ffd5bf75080e8ff5ce6b472c0cc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55a675310c8ba418e5a59beb7317e21e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/354ec8391bdd39377804ee4dab1d8f1c.png)
![](https://img.xkw.com/dksih/QBM/2021/6/29/2753469786300416/2781031193870336/STEM/37dc3ed6-8a66-441e-993b-dff3af0ce8c3.png?resizew=282)
(Ⅰ)求四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/185e2811de8461a7d5032872258bf433.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ea9995a58cbfbd0f8a5c712c2bcce4.png)
(i)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecf6c62979a7aa534a191d8387a741e8.png)
(ii)求直线CE与平面PBC所成角的正弦值.
您最近一年使用:0次
2021-08-07更新
|
1168次组卷
|
2卷引用:浙江省湖州市2020-2021学年高一下学期期末数学试题
解题方法
7 . 如图,已知在四棱锥
中,底面
是梯形,
且
,平面
平面
,
,
.
![](https://img.xkw.com/dksih/QBM/2021/7/3/2756182890135552/2780130374574080/STEM/8fbd3c9dcef645e8b67db52ac0c1a8d0.png?resizew=225)
(1)证明:
;
(2)若
,
,求四棱锥
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afc7f759828fe6a2e65e7c43070237f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b3fe90c8fbaff2d796519ce93d66f3d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae1e04eeb4de72e5750dae77bcb6f88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec3598c1f4f00f420abdeab0396391f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3196a55fe41e0c45af1a5ef8a704f4e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afbe495090bac9d0bfa5b297a63d3d12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbb05b8b630052ff544249ebd72d95d.png)
![](https://img.xkw.com/dksih/QBM/2021/7/3/2756182890135552/2780130374574080/STEM/8fbd3c9dcef645e8b67db52ac0c1a8d0.png?resizew=225)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bafa8c14100a4f847b41b9148954116c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcc1be8b5824c7cf10e237488cda10d6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2301985c94d0995eb73a4a0c6442b67e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a42fe1d6d73e8b6a650f279b0d9e872.png)
您最近一年使用:0次
2021-08-06更新
|
923次组卷
|
2卷引用:湖北省黄冈市2020-2021学年高一下学期期末数学试题
8 . 如图,在三棱锥
中,平面
平面
,
,
,若O为BC的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/26/210e1b7f-f840-48f0-b0f8-2da14a7af107.png?resizew=209)
(1)证明:
平面
;
(2)求点C到平面
的距离;
(3)设线段
上有一点M,当AM与平面
所成角的正弦值为
时,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b6e6192cf24ada791c26c2d6d434069.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee53e1960b0eb6a6779a57a855fc2921.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/26/210e1b7f-f840-48f0-b0f8-2da14a7af107.png?resizew=209)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d30637da200a07672ae231b4c5c09cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)求点C到平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9c9cfa597b444b5c9dbae7a825a695.png)
(3)设线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18e5ef91fb27dd684a27ae7f1993cfba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc9c9cfa597b444b5c9dbae7a825a695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21061c88c73e333c85933ed91e1ca393.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
您最近一年使用:0次
2020-12-30更新
|
731次组卷
|
2卷引用:江苏省南通市海安高级中学2020-2021学年高二上学期第二次阶段检测数学试题
名校
9 . 如图,在四棱锥
中,
,
,
,
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/9bd3ce4d-4392-41ba-8d97-14fbf6be5238.png?resizew=162)
(1)证明:
.
(2)若平面
平面
,经过
、
的平面
将四棱锥
分成左、右两部分的体积之比为
,求平面
与平面
所成锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5164a3cc47e266446d49127e2ef10c37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f90126f831d6600522ecaa66c2a8b9c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2ea9d3df7c2bcdf135dedd1554fb82b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88983e688ce8b02ae6237553d1226b3f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/12/9bd3ce4d-4392-41ba-8d97-14fbf6be5238.png?resizew=162)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/306b9504b52df5ad6697fa87200e8a44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5164a3cc47e266446d49127e2ef10c37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37d65e051e943ab28fa57aee2fb57994.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d78fc7fcb2762de28dcef8aa3aa0e49.png)
您最近一年使用:0次
2021-05-19更新
|
2176次组卷
|
11卷引用:河南省2021届高三仿真模拟考试数学(理科)试题
河南省2021届高三仿真模拟考试数学(理科)试题河北省沧州市2021届高三二模数学试题湖南省永州市省重点中学2021届高三下学期5月联考数学试题辽宁省朝阳市2021届高三四模考试数学试题辽宁省2021届高三5月冲刺数学试题广东省部分学校2021届高三下学期5月联考数学试题辽宁省抚顺市六校协作体2020-2021学年高三5月二模数学试题江苏省常州市新桥高级中学2021届高三下学期三模数学试题安徽省皖淮名校2020-2021学年高二下学期5月联考理科数学试题(已下线)专题04 空间向量与立体几何的压轴题(二)-【尖子生专用】2021-2022学年高二数学考点培优训练(人教A版2019选择性必修第一册)河南省信阳高级中学2022-2023学年高二上学期10月月考数学试题
名校
解题方法
10 . 七面体玩具是一种常见的儿童玩具.在几何学中,七面体是指由七个面组成的多面体,常见的七面体有六角锥、五角柱、正三角锥柱、Szilassi多面体等.在拓扑学中,共有34种拓扑结构明显差异的凸七面体,它们可以看作是由一个长方体经过简单切割而得到的.在如图所示的七面体
中,
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97809620648ce0e673acb9571de6920f.png)
①
平面
;
②
平面
;
(2)求该七面体的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8986146f5dfe7f246149773fb0ff5e89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed04b01505bbd8a4ac0bc12e46f23bf6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97809620648ce0e673acb9571de6920f.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06222ee533c2484ab25321a6abbf98cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a22d6b860f06fe23618b0d3de6768fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65277734669566578cbb7d690bb200fb.png)
(2)求该七面体的体积.
您最近一年使用:0次
2021-05-29更新
|
2249次组卷
|
9卷引用:湖北省武汉市华中师范大学第一附属中学2021届高三下学期5月高考押题卷理科数学试题
湖北省武汉市华中师范大学第一附属中学2021届高三下学期5月高考押题卷理科数学试题(已下线)湖北省武汉市华中师范大学第一附属中学2021届高三下学期5月高考押题卷文科数学试题广东省珠海市第二中学2021届考前模拟数学试题(已下线)专题10 立体几何-备战2022年高考数学(文)母题题源解密(全国乙卷)(已下线)专题35 立体几何中的探索性问题求解策略-学会解题之高三数学万能解题模板【2022版】(已下线)必刷卷02(文)-2022年高考数学考前信息必刷卷(全国甲卷)苏教版(2019) 必修第二册 过关斩将 章节测试 第13章 立体几何初步(已下线)专题3 空间几何体的体积运算(提升版)(已下线)8.6.1直线与直线垂直+8.6.2直线与平面垂直——课后作业(提升版)