名校
解题方法
1 . 为了求一个棱长为
的正四面体的体积,某同学设计如下解法.
解:构造一个棱长为1的正方体,如图1:则四面体
为棱长是
的正四面体,且有
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/6/16/6db5d8bf-a942-4eb1-b74e-0d41be5b6734.png?resizew=583)
(1)类似此解法,如图2,一个相对棱长都相等的四面体,其三组棱长分别为
,
,
,求此四面体的体积;
(2)对棱分别相等的四面体
中,
,
,
.求证:这个四面体的四个面都是锐角三角形;
(3)有4条长为2的线段和2条长为
的线段,用这6条线段作为棱且长度为
的线段不相邻,构成一个三棱锥,问
为何值时,构成三棱锥体积最大,最大值为多少?
[参考公式:三元均值不等式
及变形
,当且仅当
时取得等号]
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
解:构造一个棱长为1的正方体,如图1:则四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68ac02c2f91cadb1e328bc6ab9b9c491.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f878ffcff2ca25a434cbeea7d5c841.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/6/16/6db5d8bf-a942-4eb1-b74e-0d41be5b6734.png?resizew=583)
(1)类似此解法,如图2,一个相对棱长都相等的四面体,其三组棱长分别为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50690dab38f4512eb72e18b7f86cf6f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4056761b8f826eeb6ad8c9a151d3c9c.png)
(2)对棱分别相等的四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c220eadc312101e2fb89dfe920f7b30d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7de966c316db1013defc56372fcf814e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8d2530e7023b2345c651e8f53629ff1.png)
(3)有4条长为2的线段和2条长为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
[参考公式:三元均值不等式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ffb6b373d2e672bb2afc8de547861a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4849ff71159df2bb9099b26065d81e1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44acc0ee22dc4b7750e8be825e7c1355.png)
您最近一年使用:0次
2021-07-15更新
|
814次组卷
|
2卷引用:重庆市西南大学附属中学2020-2021学年高一下学期期末数学试题
名校
解题方法
2 . 某校学生利用解三角形有关知识进行数学实践活动.
处有一栋大楼,某学生选
、
(与
在同一水平面上)两处作为测量点,测得
的距离为
,
,
,在
处测得大楼(大楼
与水平面
垂直)楼顶
的仰角
为
.
两点间的距离;
(2)求大楼的高度及二面角
的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c56c87fd6bf8a44244ba51a9d244e22.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c42ed2e5bd5a0f033e24008697bf4963.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d0ec7f8ab857cd441a82389b246230a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d822262ff00915910e5b87d81ad1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.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/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a3e7d1f1890863b7483a98b618119.png)
(2)求大楼的高度及二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab6d47edbcc2ae6efcfd7f28e401e3e9.png)
您最近一年使用:0次
名校
3 . 如图,在堑堵
中(注:堑堵是一长方体沿不在同一面上的相对两棱斜解所得的几何体,即两底面为直角三角形的直三棱柱,最早的文字记载见于《九章算术》商功章),已知
平面
,
,
,点
、
分别是线段
、
的中点.
平面
;
(2)求直线
与平面
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b40ed494a8aa0304858a5f6919ac2ae5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ccc0d735dc14c92a8926d9b1def6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6f8e9ec412ea0355e4e5cd06c60e5fee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f90e17995e2f71e297d94ae51c7e5b1f.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/fa6cb992b6faad4744f85d73a3b76dd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7592c4f01c8e06c7ee90df5b9413a9f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e168672b47d7e64dc1b404f8882c7dcf.png)
您最近一年使用:0次
2023-08-02更新
|
943次组卷
|
7卷引用:浙江省宁波市慈溪市2022-2023学年高一下学期期末数学试题
浙江省宁波市慈溪市2022-2023学年高一下学期期末数学试题辽宁省朝阳市建平县实验中学2023-2024学年高二上学期期中数学试题(已下线)压轴题立体几何新定义题(九省联考第19题模式)练(已下线)第四章 立体几何解题通法 专题二 升维法 微点3 升维法综合训练【培优版】(已下线)第六章 突破立体几何创新问题 专题一 交汇中国古代文化 微点1 与中国古代文化遗产有关的立体几何问题(一)【基础版】(已下线)重难点专题13 轻松搞定线面角问题-【帮课堂】(苏教版2019必修第二册)(已下线)专题08立体几何期末14种常考题型归类(1)-期末真题分类汇编(人教B版2019必修第四册)
名校
4 . 国家主席习近平指出:中国优秀传统文化有着丰富的哲学思想、人文精神、教化思想、道德理念等,可以为人们认识和改造世界提供有益启迪.我们要善于把弘扬优秀传统文化和发展现实文化有机统一起来,在继承中发展,在发展中继承.《九章算术》作为中国古代数学专著之一,在其“商功”篇内记载:“斜解立方,得两壍堵.斜解壍堵,其一为阳马,一为鳖臑”.刘徽注解为:“此术臑者,背节也,或曰半阳马,其形有似鳖肘,故以名云”.鳖臑,是我国古代数学对四个面均为直角三角形的四面体的统称.在四面体
中,
平面
.
![](https://img.xkw.com/dksih/QBM/2021/7/9/2760356260454400/2761304616468480/STEM/acdd4b31b8504295b1d47e95e3567ddf.png?resizew=455)
(1)如图1,若
、
、
分别是
、
、
三边的的中点,
在
上,且
,求证:
平面
;
(2)如图2,若
,垂足为
,且
,
,
,求直线
与平面
所成角的大小;
(3)如图2,若平面
平面
,求证:四面体
为鳖臑.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9e1bf2cc650448488a19c6301125b31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cf468f5132e14ee1d8cc766808b11af.png)
![](https://img.xkw.com/dksih/QBM/2021/7/9/2760356260454400/2761304616468480/STEM/acdd4b31b8504295b1d47e95e3567ddf.png?resizew=455)
(1)如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dec0d25fa88ea9f0b003b83b7e2fe88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f28f9f503c0a023ed7e78e48123cc95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d900531973c546625694146fa1509ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/618c5704137191d21172232bdb26b4d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68b40d0d2f3cdd8981bb792ad87efb42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71807a35b3170fce28ee6edf4c00d083.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf4c26f3f4d96117f087400a0f32ece8.png)
(3)如图2,若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1095b030f441de5fb223781b00f3dd9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aeb5255e2159617505e0c87d01437a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9e1bf2cc650448488a19c6301125b31.png)
您最近一年使用:0次
2021-07-10更新
|
391次组卷
|
2卷引用:四川省成都市第七中学2020-2021学年高一下学期期末考试数学试题