1 . 如图所示,图(1)中的
中,
,
,
是
的中点,现将
沿
折起,使点
到达点
的位置,且满足
,得到如图(2)所示的三棱锥
,点
、
分别是棱
、
的中点,
、
分别在棱
、
上,满足
,
.
![](https://img.xkw.com/dksih/QBM/2022/3/24/2943275079565312/2945845829574656/STEM/6ecebc91-3eef-4c0b-8fce-ab3ef3ab4ae3.png?resizew=293)
(1)求证:
平面
;
(2)求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aedf65d7d930fdb972d4802c0dea8b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f121eabff3c62c1a196d9ca5f6f83f0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.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/6ed66431681da1db8f7cb0f40cd19201.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08a9ec3b527947cad9caa4537e0cb7e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.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/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c971d210a955c4c4be75c38f34a68fe9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2034ad1ac2d047eba16071d9d470cbfa.png)
![](https://img.xkw.com/dksih/QBM/2022/3/24/2943275079565312/2945845829574656/STEM/6ecebc91-3eef-4c0b-8fce-ab3ef3ab4ae3.png?resizew=293)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f9d682e5d3cc8573574d8d11636758.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbae23711fb470b75778130c91e0e15.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
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2 . 筝形是指有一条对角线所在直线为对称轴的四边形.如图,四边形
是一个筝形,
,
,
,沿对角线
将
折起到
点,形成四棱锥
.
![](https://img.xkw.com/dksih/QBM/2021/12/28/2882139470028800/2886327225851904/STEM/451a4bb5-5624-4cf1-be10-58930b4f4b72.png?resizew=469)
(1)点
为线段
中点,求证:
平面
;
(2)当
时,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dc13307a688f0820d9bc8b946b04b7e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c1b292baa30ffc34df3a47d57b60c0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e75fb6f9ad72dea327a6895915cd5355.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://img.xkw.com/dksih/QBM/2021/12/28/2882139470028800/2886327225851904/STEM/451a4bb5-5624-4cf1-be10-58930b4f4b72.png?resizew=469)
(1)点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f369bec2d5682bf6b8b317a08aff546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2977ae4bfa32de8c6f0fb136205c4fe7.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c9c9da1307b4cd926a5f5cdda2d4845.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2194add18a7df1a23cf1554dc2da1b40.png)
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2022-01-03更新
|
942次组卷
|
6卷引用:浙江省绍兴市诸暨市海亮高级中学2021-2022学年高三上学期12月选考数学试题
浙江省绍兴市诸暨市海亮高级中学2021-2022学年高三上学期12月选考数学试题(已下线)专题10 立体几何线面位置关系及空间角的计算(练)--第一篇 热点、难点突破篇-《2022年高考数学二轮复习讲练测(浙江专用)》(已下线)解密12 空间向量在空间几何体的应用(讲义)-【高频考点解密】2022年高考数学二轮复习讲义+分层训练(新高考专用)湖南省2022届高三下学期3月调研考试数学试题浙江省绍兴市诸暨市海亮高级中学2022届高三下学期高考前最后一卷数学试题山东省日照实验高级中学2023-2024学年高二上学期第一次阶段性考试数学试题
3 . 如图,在
中,
,斜边
,现将
绕AC旋转一周得到一个圆锥,BD为底面圆的直径,点P为圆锥的内切球O与CD的切点,
为圆锥底面圆周上异于B,D的一点.
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899096901861376/2909066567385088/STEM/1d4def76-13cf-4a3e-b978-c387907a7275.png?resizew=259)
(1)求证:
平面
;
(2)当
时,求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/564e238659b316ab3275829470ce0c89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e65a3e478bb87d094e3a0af30dd10ae8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://img.xkw.com/dksih/QBM/2022/1/21/2899096901861376/2909066567385088/STEM/1d4def76-13cf-4a3e-b978-c387907a7275.png?resizew=259)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/373f735f0f04d11f1951eaef1bb78b6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/113c87d7b997847259f17ee8576ee44c.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/295aced98768ce261e00fe6660a427a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9655de237e97442f6920fad144b2d85b.png)
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4 . 如图,P为圆锥的顶点,O为圆锥底面的圆心,圆锥的底面直径
,母线
,M是PB的中点,四边形OBCH为正方形.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/10/826e893f-c0dc-424b-b5d2-f3bc09b306ee.png?resizew=278)
(1)设平面
平面
,证明:
;
(2)设D为OH的中点,N是线段CD上的一个点,当MN与平面PAB所成角最大时,求MN的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91758aec8a689f6a857ea3c289378680.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/10/826e893f-c0dc-424b-b5d2-f3bc09b306ee.png?resizew=278)
(1)设平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec0b3e4c789528fe60ac9b8ce990bc96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3e61111c1e9b98b79615f75540175c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcfa68f5a67f2546c4e6688504e490e.png)
(2)设D为OH的中点,N是线段CD上的一个点,当MN与平面PAB所成角最大时,求MN的长.
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2022-07-22更新
|
4307次组卷
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9卷引用:山东省青岛市2022届高三下学期5月二模考试数学试题
山东省青岛市2022届高三下学期5月二模考试数学试题(已下线)专题24 立体几何解答题最全归纳总结-1江苏省扬州市宝应县2022-2023学年高三上学期期初检测数学试题湖北省襄阳市第四中学2022-2023学年高二上学期新起点考试数学试题湖北省武汉市部分重点中学2022-2023学年高二上学期10月联考数学试题(已下线)专题16 空间向量及其应用(模拟练)江苏省盐城市2022-2023学年高三上学期11月阶段考试数学试题(已下线)第3章 空间向量及其应用(基础、常考、易错、压轴)分类专项训练(原卷版)山东省烟台市爱华高级中学2023-2024学年高二上学期期中考试数学试题
解题方法
5 . 如图,旗杆AB高8m,它的顶端A挂着两条长10m的绳子,拉紧绳子,使绳子的末端分别与地面接触,记接触点为C,D(和旗杆脚B不在同一条直线上).
![](https://img.xkw.com/dksih/QBM/2021/12/24/2879279051956224/2882333312786432/STEM/3bc99a29795d44538ecbca4e208cf463.png?resizew=260)
(1)如果C,D两点和旗杆脚B的距离都是6m,就证明旗杆和地面垂直,请写出证明过程;
(2)如果E为绳子AC的中点,在旗杆AB上是否存在一点F,使EF和地面平行?如果存在,请确定点F的位置,并写出证明过程;如果不存在,请说明理由.
![](https://img.xkw.com/dksih/QBM/2021/12/24/2879279051956224/2882333312786432/STEM/3bc99a29795d44538ecbca4e208cf463.png?resizew=260)
(1)如果C,D两点和旗杆脚B的距离都是6m,就证明旗杆和地面垂直,请写出证明过程;
(2)如果E为绳子AC的中点,在旗杆AB上是否存在一点F,使EF和地面平行?如果存在,请确定点F的位置,并写出证明过程;如果不存在,请说明理由.
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6 . 如图,已知正四棱锥
与正四面体
所有的棱长均为
.
![](https://img.xkw.com/dksih/QBM/2021/7/8/2759751319707648/2777760744325120/STEM/ede128cb-f7cd-45e0-b834-efbf130c4aef.png?resizew=477)
(1)若
为
的中点,证明:
平面
;
(2)把正四面体
与正四棱锥
全等的两个面重合,排成一个新的几何体,问该几何体由多少个面组成?并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/858d5bfe390d0cb79cee200241240a31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://img.xkw.com/dksih/QBM/2021/7/8/2759751319707648/2777760744325120/STEM/ede128cb-f7cd-45e0-b834-efbf130c4aef.png?resizew=477)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d4db9b82b67efe45a02fca32bfcf5dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca6d50356a01ae13936f1bd8efa94c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb304d905125170bebfada27e7ed8960.png)
(2)把正四面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/858d5bfe390d0cb79cee200241240a31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
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2021-08-02更新
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892次组卷
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3卷引用:福建省福州市2020-2021学年高一下学期期末数学试题
7 . 在如图所示的圆台中,
是圆台的轴截面,
,
分别是上、下底面圆的圆心,
是下底面圆周上异于
,
的一点,设圆台的上、下底而圆的半径分别为
与
,高为
,体积为
.
![](https://img.xkw.com/dksih/QBM/2021/7/9/2760252207202304/2790354858409984/STEM/67614575-75d5-451e-9825-897e7386b609.png?resizew=306)
(1)若
,
外别是
与
的中点,试判断直线
与平面
的位置关系,并加以证明;
(2)若
,求二面角
的正切值;
(3)在估测圆台的体积时,常用近似公式
来计算,其中圆台的中截面是指与上、下两个底而平行,且到两个底面距离相等的截而,试判断与
与
的大小关系,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](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/11bc05f41215f9894e11d1df0465751a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d33602cdebfa2ce11913f13da2bd861.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
![](https://img.xkw.com/dksih/QBM/2021/7/9/2760252207202304/2790354858409984/STEM/67614575-75d5-451e-9825-897e7386b609.png?resizew=306)
(1)若
![](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/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c09afc70f448545336304333d5b5658b.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5333d85c95349bb62729f7ac91e81fc0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5617a404c5a3356753136e5a6b6d51e5.png)
(3)在估测圆台的体积时,常用近似公式
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb55d3a56d4c7fa5127e11e463d9fcf8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212da8a37577d441928d41770e982e44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be54e84508decfcce6d2fcbe6c8c1a92.png)
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