2024高三·全国·专题练习
1 . 作出过三点的截面,其中
为所在棱上中点(三条边都在正方体内部).
(1)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/20/be1032c4-5a36-45fa-b129-fb79dd0db354.png?resizew=168)
(2)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/20/02ec717e-6dd1-4fd2-96df-49d4e5d5db8a.png?resizew=168)
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2 . 上海中心大厦是上海市的地标建筑,现为中国第一高楼.为有效减少建筑所受的风荷载,通常对建筑体型进行一定的扭转.上海中心大厦的主楼可近似看成将正三棱柱的一个底面扭转所得的几何体;将正三棱柱
的底面
在其所在平面内绕
的中心逆时针旋转
得到
,再分别连接
、
、
、
、
、
所得的几何体.已知大厦的主楼高度约为
米,底层面积(即
的面积)约为
平方米.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/31/fdf3aa87-7bc9-4fc4-a1b2-d692595b7966.png?resizew=149)
(1)求证:
;
(2)试分别以正三棱柱
和几何体
为模型估算大厦主楼的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d5bca00fa20e6e80480b9d06d2e52ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8ee6e1d480ece7117e1f87ebf4bbeea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f986a0d8f37177dcccfee3898a66fd00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20c431cd12f858f0bc8dabb1d8c0b8e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/020ebe1219437129358b986eb9e70bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/300d29bf2277a510ab443c1e2a55e1bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4336253885d52e43ba6eaa297ea847b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3157362e4455a2176539f8bdcfcea93c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2faca11afa8ddaa19cde2e91ee5983f7.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/31/d2649820-fc38-45b9-ba26-9032c8bf3c25.jpg?resizew=128)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/31/fdf3aa87-7bc9-4fc4-a1b2-d692595b7966.png?resizew=149)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/154c43b58f7f6389d6d71aa520b6c34f.png)
(2)试分别以正三棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71f2185273bf04c11118c7954f7ec822.png)
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3 . 在空间中,下列说法错误的是( )
A.过直线外一点作已知直线的垂线有无数条 |
B.两条平行直线中的一条平行于一个平面,则另一条也一定平行于该平面 |
C.一条直线分别与两个相交平面平行,那么该直线一定与两平面的交线平行 |
D.两个平面垂直,过其中一个平面内的一点作另一个平面的垂线有且只有一条 |
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解题方法
4 . 在四面体
中,
分别是棱
的中点,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42c2d86d8daea5e652d99fe1c6bc3f9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/89c9f0b87f497c68422134539c2e60a4.png)
A.![]() |
B.若![]() ![]() |
C.若![]() ![]() |
D.若![]() ![]() |
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解题方法
5 . 若平面
的斜线l在
内的射影为
,直线
,且
,则b与l( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13dea1bd3d0dd84b8b6f6ff634c5600c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b330d69a949d9b55f4b6f18f47e0a37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4751b438a4abc833bcb1909dd7d82027.png)
A.必相交 | B.必为异面直线 | C.垂直 | D.平行 |
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解题方法
6 . 如图,已知CD是异面直线CA、DB的公垂线.
,垂足为A;
,垂足为B.
.求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da70d2d84c096a431e03a5ad3fad76d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7111b543f2f2641a9eeef17596bcd2fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fb4b89a738f9163c9b437d219eeb29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c11e72609ba0fbefe03c9f24165cbf11.png)
![](https://img.xkw.com/dksih/QBM/2022/4/20/2962142131658752/2962830652391424/STEM/5c488868-2db9-4f5a-9bac-65966632e9c9.png?resizew=154)
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2022高三·全国·专题练习
解题方法
7 . 如图,已知底面为平行四边形的四棱锥
中,平面
与直线
和直线
平行,点
为
的中点,点
在
上,且
.
![](https://img.xkw.com/dksih/QBM/2022/1/9/2890852478820352/2893807159459840/STEM/a9ad9438-b015-4bbb-a063-31226cf99d0c.png?resizew=184)
(1)求证:四边形
是平行四边形;
(2)求作过
作四棱锥
的截面,使
与截面平行(写出作图过程,不要求证明).截面的定义:用一个平面去截一个几何体,平面与几何体的表面的交线围成的平面图形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a315e4b5963127bf8550cde03ca1966d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d5ae8d1e8f3edf571aeca89135dcff3.png)
![](https://img.xkw.com/dksih/QBM/2022/1/9/2890852478820352/2893807159459840/STEM/a9ad9438-b015-4bbb-a063-31226cf99d0c.png?resizew=184)
(1)求证:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a315e4b5963127bf8550cde03ca1966d.png)
(2)求作过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
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解题方法
8 . 下列命题正确的是( )
A.与平面内无数条直线垂直的直线与该平面垂直 |
B.过直线外一点可以作无数条直线与该直线平行 |
C.正四面体的外接球球心和内切球球心恰好重合 |
D.各面都是等腰三角形的三棱锥一定是正三棱锥 |
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2021-10-07更新
|
859次组卷
|
3卷引用:“星云”2022届高三上学期第二次线上联考数学试题
9 . 已知直线
且相距28
,
在
、
所确定的平面
外,
且相距17
,
和平面
相距15
,求
与
间的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e41d3f7d55fcbaebc4e2450ac63a3dc5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
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解题方法
10 . 如图,在四棱锥F-ABCD和四面体
中,四边形ABCD为矩形,两个△FAD和△
全等,△
为等边三角形,且
,棱锥F-ABCD的四条侧棱相等,
⊥平面
,现将两个几何体中的△FAD和△
重合,构成一个新的几何体FEABCD,如图(2),并且CD⊥EA.
![](https://img.xkw.com/dksih/QBM/2021/5/20/2725172812292096/2752124525174784/STEM/80032bcc785144278efacdfac559cf2a.png?resizew=523)
(1)证明∶点E为两个平面BAF和平面CDF的一个公共点;
(2)求平面AED与平面BCF所成角(锐角)的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c3772fa9c8b48285a9808abe6b3ff54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9c9c1dde743d970460ba1e9b7de8739.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49dcaa6503841b56e96b8e4a3660f816.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a74abf067e11e6a83956badea7c40974.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df34db81c4111adec4d81540e2b003f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49dcaa6503841b56e96b8e4a3660f816.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9c9c1dde743d970460ba1e9b7de8739.png)
![](https://img.xkw.com/dksih/QBM/2021/5/20/2725172812292096/2752124525174784/STEM/80032bcc785144278efacdfac559cf2a.png?resizew=523)
(1)证明∶点E为两个平面BAF和平面CDF的一个公共点;
(2)求平面AED与平面BCF所成角(锐角)的余弦值.
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