名校
1 . 立德中学积极开展社团活动,在一次社团活动过程中,一个数学兴趣小组发现《九章算术》中提到了“刍甍(méng)”这个五面体,于是他们仿照该模型设计了一道数学探究题,如图1,
分别是边长为4的正方形三边
的中点,先沿着虚线段
将等腰直角三角形
裁掉,再将剩下的五边形
沿着线段
折起,连接
就得到了一个“刍甍”(如图2).
![](https://img.xkw.com/dksih/QBM/2022/12/12/3129325021765632/3129834832379904/STEM/900f6d97b77f4452a4c9ead8dd3cdcd4.png?resizew=423)
(1)若
是四边形
对角线的交点,求证:
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7090ad13cf3664c89cdb2288779a9669.png)
(2)若二面角
的大小为
,求直线
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6241f9ef86cd0a902cbadaf336767dbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe723f84ba0818b496df2a414cc959a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45f70627e259fa4e67edff13bb3b4d9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d4c6641b74b01218e302370ebf71131.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/654830d1b3b2dc3c6ffcf3654e1d8ac0.png)
![](https://img.xkw.com/dksih/QBM/2022/12/12/3129325021765632/3129834832379904/STEM/900f6d97b77f4452a4c9ead8dd3cdcd4.png?resizew=423)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e826b8202fa0e17245dcc68426c923a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d786346b0e3f2d6666a2e7bf0b7e1251.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7090ad13cf3664c89cdb2288779a9669.png)
(2)若二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/770d42343599d3f26f0e0de8d5849f52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/732250efe9c8c0cbca127fb2ed2a4bf9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7090ad13cf3664c89cdb2288779a9669.png)
您最近一年使用:0次
2022-12-13更新
|
1183次组卷
|
21卷引用:甘肃省兰州市西固区兰州市第六十一中学2023届高三上学期期末理科数学试题
甘肃省兰州市西固区兰州市第六十一中学2023届高三上学期期末理科数学试题江苏省无锡市江阴市2022届高三下学期最后一卷数学试题(已下线)第一章 空间向量与立体几何综合测试-2022年暑假高一升高二数学衔接知识自学讲义(人教A版2019)(已下线)专题08 立体几何综合-备战2023年高考数学母题题源解密(新高考卷)空间向量的应用空间向量与立体几何中的高考新题型湖北省潜江市园林高级中学2022-2023学年高二上学期10月月考数学试题山东省潍坊市昌邑市潍坊实验中学2022-2023学年高二上学期10月月考数学试题山东省济宁市梁山县第一中学2022-2023学年高二上学期10月月考数学试题湖北省武汉市第二十中学2022-2023学年高二上学期10月月考数学试题湖南省张家界市慈利县第一中学2022-2023学年高三上学期第四次月考数学试题湖北省随州市曾都区第一中学2022-2023学年高二上学期期末模拟数学试题江西省新余市2023届高三上学期期末质量检测数学(理)试题河北省唐山市开滦第二中学2023届高三上学期第四次线上考试数学试题湖南省长沙市浏阳市2022-2023学年高二上学期期末数学试题湖北省黄冈市浠水县第一中学2022-2023学年高二下学期3月质量检测数学试题山东省日照市实验高级中学2022-2023学年高二上学期第一次阶段(10月月考)数学试题山西大学附属中学校2023-2024学年高二上学期10月模块诊断数学试题四川省眉山市仁寿第一中学校南校区2023-2024学年高二上学期第一次质量检测数学试题湖南省张家界市慈利县第一中学2022-2023学年高二上学期第四次月考数学试题(已下线)第六章 突破立体几何创新问题 专题二 融合科技、社会热点 微点3 融合科技、社会热点等现代文化的立体几何和问题综合训练【培优版】
名校
2 . 如图,四棱锥P-ABCD中,AP⊥平面PCD,
,
,
,E为AD的中点,AC与BE相交于点O.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/23/1d2c665d-7284-4e30-920e-8ac8b13b24ea.png?resizew=222)
(1)求证:PO⊥平面ABCD;
(2)求直线AB与平面PBD所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b377f22aafd3742ad860f77abaacef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f2d58e450193da0539a687dabf0bfa.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/23/1d2c665d-7284-4e30-920e-8ac8b13b24ea.png?resizew=222)
(1)求证:PO⊥平面ABCD;
(2)求直线AB与平面PBD所成角的正弦值.
您最近一年使用:0次
名校
3 . 如图,在四棱锥P-ABCD中,平面
平面ABCD,PA=PD,
,
,AD=CD=2,AB=3,E是棱AD的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/080cffce-e033-4c60-b855-93295692faf0.png?resizew=165)
(1)证明:
平面PCE;
(2)若
,求平面PCE与平面PAB所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10df84d553a8826a7ce9bff4bf0d95b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4ab7e657f01bdfa235f8c4d6681d13.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/20/080cffce-e033-4c60-b855-93295692faf0.png?resizew=165)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2ffc6952e988d04f22f0fb2f7f0ab7b.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3ad66112b09c909cab417085702ec00.png)
您最近一年使用:0次
2022-11-19更新
|
392次组卷
|
4卷引用:甘肃省兰州市兰州西北中学2022-2023学年高三上学期期中数学(理科)试题
名校
4 . 在多面体
中,平面
平面ABCD,EDCF是面积为
的矩形,
,
,
.
![](https://img.xkw.com/dksih/QBM/2022/8/26/3052759954268160/3053455803826176/STEM/4563ea97675a45c8bdfc6abaffa26254.png?resizew=203)
(1)证明:
.
(2)求平面EDCF与平面EAB夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2fc1129846f37afdafd751627c450d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15ea464a0929a33bedd2ee95cdb66ba8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3402ea855e2ae2dcd98f607bef4fdd6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a75b1354b8b783a65ee5e3bc596a976.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://img.xkw.com/dksih/QBM/2022/8/26/3052759954268160/3053455803826176/STEM/4563ea97675a45c8bdfc6abaffa26254.png?resizew=203)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cade9d1bac990f2014ff8310613e2613.png)
(2)求平面EDCF与平面EAB夹角的余弦值.
您最近一年使用:0次
2022-08-27更新
|
453次组卷
|
7卷引用:甘肃省兰州市西固区兰州市第六十一中学2023届高三上学期10月月考理科数学试题
名校
解题方法
5 . 如图所示,在直四棱柱
中,底面ABCD是等腰梯形,
,
,
,四边形
是正方形.
![](https://img.xkw.com/dksih/QBM/2022/5/24/2986420541005824/2987744622305280/STEM/0babbae5-66e1-4454-955f-aeecba309a60.png?resizew=245)
(1)指出棱
与平面
的交点E的位置(无需证明),并在图中将平面
截该四棱柱所得的截面补充完整;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fff774b4b0087a6f304ce930d359be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12143a06ed24558d8cc7ad39961d3e1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f945a69cf7e8213e50622125cde652f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b724168afaee2ecddf97257180be18.png)
![](https://img.xkw.com/dksih/QBM/2022/5/24/2986420541005824/2987744622305280/STEM/0babbae5-66e1-4454-955f-aeecba309a60.png?resizew=245)
(1)指出棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b2213b575a7cfaffcdf91885005c7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b2213b575a7cfaffcdf91885005c7d.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8375f8f2a0dd3e0212ce52d334952c.png)
您最近一年使用:0次
2022-05-26更新
|
752次组卷
|
4卷引用:甘肃省兰州市第六十一中学2022-2023学年高三上学期11月期中考试理科数学试题
名校
解题方法
6 . 在四棱锥
中,
底面
,底面
是边长为2的菱形,
,
是
的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/25/46f6f48b-a926-440d-9334-c6c4faf33ea1.png?resizew=159)
(1)求证:平面
平面
;
(2)直线
与平面
所成角为
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea4f5eec0addba78f2e0cdfb7ecc59a6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/25/46f6f48b-a926-440d-9334-c6c4faf33ea1.png?resizew=159)
(1)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a26a7784c7419d8359fb119c8ecc03d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
(2)直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79a97bb4dcfab4ec7539bc783d563c49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/334b0be972ebf5a46333c0c4369aa90a.png)
您最近一年使用:0次
2023-02-24更新
|
777次组卷
|
8卷引用:甘肃省兰州第一中学2022-2023学年高二下学期期中数学试题
名校
7 . 已知四棱锥
中,底面
为菱形,点E为校PC上一点(与P、C不重合),点M、N分别在棱PD、PB上,平面
平面
.
![](https://img.xkw.com/dksih/QBM/2022/4/18/2960912373121024/2968004299522048/STEM/782e368afb0548e7bcb0517f95a8e8d4.png?resizew=245)
(1)求证:
平面
;
(2)若
为
中点,
,
,
,求二面角
的正弦值.
![](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/b997c9c4f053b99b47d0307cdf42516a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://img.xkw.com/dksih/QBM/2022/4/18/2960912373121024/2968004299522048/STEM/782e368afb0548e7bcb0517f95a8e8d4.png?resizew=245)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e3c0430146b7b8d40ebb721a4d0de19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1885efcff0b903e314057dd153578600.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b17e228085eaa3c91d68620582ab6b63.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45884ed34b6a258ee31c137f13b01610.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e4125524caac016727c80d2722c5ba3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a367662d767d6fe559a55159ff24d855.png)
您最近一年使用:0次
2022-04-28更新
|
702次组卷
|
3卷引用:甘肃省兰州市2022届高三诊断考试理科数学试题
甘肃省兰州市2022届高三诊断考试理科数学试题(已下线)考点17 点、直线、平面之间的位置关系-2-(核心考点讲与练)-2023年高考数学一轮复习核心考点讲与练(新高考专用)江西省贵溪市实验中学2024届高三上学期11月第二次模拟检测数学试题
名校
8 . 如图,在多面体ABCDEF中,四边形ABCD为直角梯形,
,AB⊥AD,四边形ADEF为正方形,平面ADEF⊥平面ABCD.BC=3AB=3AD,M为线段BD的中点.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/2/e27ee0cd-1c32-4e61-82e3-d56b0f8dfcd6.png?resizew=273)
(1)求证:BD⊥平面AFM;
(2)求平面AFM与平面ACE所成的锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/2/e27ee0cd-1c32-4e61-82e3-d56b0f8dfcd6.png?resizew=273)
(1)求证:BD⊥平面AFM;
(2)求平面AFM与平面ACE所成的锐二面角的余弦值.
您最近一年使用:0次
2023-01-15更新
|
479次组卷
|
4卷引用:甘肃省兰州市第五十中学2022-2023学年高三第一次模拟考试数学(理科)试题
名校
9 . 在如图所示的多面体中,点
在矩形
的同侧,直线
平面
,平面
平面
,且
为等边三角形,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/78fb6e26-f2a4-4548-8bb7-2c4ce6bf1684.png?resizew=144)
(1)证明:
;
(2)求平面
与平面
所成锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7a38e6c6dfde2b19b6b47f35a439a06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd2d28f1e7a6b17401c19c34beddcbe0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6830ebecddbd9759be626289c408e4f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3521fa4923fb2e492e0b9d8c080215a5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/14/78fb6e26-f2a4-4548-8bb7-2c4ce6bf1684.png?resizew=144)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1384ffba86ff08ce9e783d5d1bc51686.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20af148464904e21f4374cc8fb886fba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f636f76d550dfb593a25eb680cff556.png)
您最近一年使用:0次
2022-02-25更新
|
1391次组卷
|
4卷引用:甘肃省兰州市第五十中学2022-2023学年高三下学期开学摸底考试数学(理科)试题
名校
10 . 如图,三棱柱
中,底面
为等腰直角三角形,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/27/50611a26-7b26-4464-9880-a65c4379bf7f.png?resizew=129)
(1)证明:
;
(2)若
,求
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e8f93af0e5bb540c2b9af4455ee395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54fdea2fb34b89726ea6b5d215e0919d.png)
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(2)若
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24bb49fdc6b6bbb2449fdf8a0de769d3.png)
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2022-12-26更新
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4卷引用:【全国百强校】甘肃省兰州第一中学2018-2019学年高二上学期期末考试数学(理)试题
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