解题方法
1 . 如图所示正四棱锥
,
,
,
为侧棱
上的点,且
.求:
的表面积;
(2)若
为
的中点,求平面
与平面
所成的二面角的余弦值;
(3)侧棱
上是否存在一点
,使得
平面
.若存在,求
的值;若不存在,试说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c56f3d2a9c05c3a13330af7a261889.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21ea52361458ce2e49ed0fe99d8e6c02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/defa5b53043ae802bb1af7d14374406d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b2883beed42e46f8f379b02ea3b68b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a6e2867f32d3f1c3cd36cd3a11a8580.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f239fbcc58fc15535db4b5084c4f7253.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(3)侧棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c2bc5e50b8dfa02601c70822252854a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c372d059202ec388960b125d4a87dc84.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fea0808c7df5a3fa6678ee5406b35b25.png)
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解题方法
2 . 如图,在空间直角坐标系
中,四棱柱
为长方体,
,点
为
的中点,
与平面
的夹角的正弦值;
(2)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c76ee17afa983fa795545b5568b80089.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce34c05c1445e027e9fc009907046e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66fb71b75b63594ebeeeebd1963eed5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/923189afc198d153c79059a827f63c87.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1c8b9d5680c06f9e28c311d67cfadd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/923189afc198d153c79059a827f63c87.png)
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解题方法
3 . 如图所示,可以将钟楼看作一个长方体,四个侧面各有一个大钟,则从8:00到10:00这段时间内,相邻两面钟的分针所成角为
的次数为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
A.2 | B.4 | C.6 | D.8 |
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4 . 如图,三棱锥
中,正三角形
所在平面与平面
垂直,
为
的中点,
是
的重心,
,
,
.
∥平面
;
(2)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0628681907ac8d7fdb94d8bc1b15feb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c4d1ecb7544d0be8669d1b19f60c4da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080db3af81b29ed10144a1c2e2a4fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33c99fc553ecb3e2afdb7058201bc642.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cbae23711fb470b75778130c91e0e15.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
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解题方法
5 . 如图,等腰梯形
中,
,
,现以
为折痕把
折起,使点
到达点
的位置,且
.
(1)证明:平面
平面
;
(2)若
为
上的一点,点
到平面
的距离为
,求二面角
的余弦值.
![](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/3988972c6cf76974243aa31e318f20f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b44f4120c94cb7176dc31fcac387b32e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/17/2bd5a4f1-1bc6-4de1-8938-e5fb470ba7d4.png?resizew=328)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d077f6da8b2c00b152d4679aa2ed7f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af68a7bf0da4f7c6f739d2e2461ad9b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9868f77d5ab5073b6145f1c6d272122e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/698335f4880c7a298f4898c83b6562bf.png)
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解题方法
6 . 如图.已知平行六面体
的底面是菱形,
,
.
;
(2)求点
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4660d993eda4d53b99bbe06a20344462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/920a7dec798f1fd44f5830504ab5bdec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02a0e00113872f921116b6c0c3177d0f.png)
(2)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/239198e40085b7dcffbe747c9c265a05.png)
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解题方法
7 . 如图所示,在四棱锥
中,
平面
,底面
是正方形,
是
的中点,
在线段
上,且
.![](https://staticzujuan.xkw.com/quesimg/Upload/formula/901c9e0d6c246c195b65100f3f622899.png)
(2)求平面
与平面
所夹二面角余弦值.
![](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/35ec0ad27763bdc075e880c9f089d305.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71bf1a9310841cd256bdb3e7fb084135.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/901c9e0d6c246c195b65100f3f622899.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212a67f115d1cbe69f100b489babe5f8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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解题方法
8 . 在棱长为1的正方体
中,点F是棱
的中点,P是正方体表面上的一点,若
,则线段
长度的最大值为________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ffae8b04ebf3c711c96d5d41dfa1420.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c798204bbe306b3efd5bc9eae594c171.png)
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解题方法
9 . 在棱长为2的正方体
中,
分别是
的中点,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1797b6b52b9ef3d5a39cfc583bafac80.png)
A.直线![]() ![]() |
B.过点![]() ![]() |
C.三棱锥![]() ![]() |
D.点![]() ![]() ![]() |
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解题方法
10 . 在棱长为2的正方体
中,E,F,M,N分别为
,
,
,
中点.
(1)求证:
平面
;
(2)求直线
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e619f087b6b7ab764362b8b64b220cd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc65c549059934e69355d8ecc245da57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f257d6a77d394ddca1f825559aadd5be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3225b8916372c7e0e4d7b71b26571e8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/6/17/8ea83e69-f4b4-44da-a585-6110ad87a320.png?resizew=166)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/073a88b42836fb88433679932b48ad03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72bdc4f7de61cf83503ccb8a81b36c47.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0215e13a9fb5574d5194aeb9507a98aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6b686178c52fdf7ac270e75c0795417.png)
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