解题方法
1 . 下图是一个正三棱柱(以
为底面)被一平面所截得到的几何体,截面为
.已知
,
,
,
.
![](https://img.xkw.com/dksih/QBM/2016/1/22/1572463379816448/1572463385960448/STEM/b721e6e1d1374f23af0df68a1c9eba5c.png)
(1)设点
是
的中点,证明:
平面
;
(2)求
与平面
所成的角的正弦值;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b16cff607cdc2d69afc70dc778acbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://img.xkw.com/dksih/QBM/2016/1/22/1572463379816448/1572463385960448/STEM/eeea238797af4d30af00fbd8270f70d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e55a2310cbba5e050488cd9296eb195d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41fd676c41d2d644928f014b0fea4689.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c122ca7141c43c15c783968f5f0dbc19.png)
![](https://img.xkw.com/dksih/QBM/2016/1/22/1572463379816448/1572463385960448/STEM/b721e6e1d1374f23af0df68a1c9eba5c.png)
(1)设点
![](https://img.xkw.com/dksih/QBM/2016/1/22/1572463379816448/1572463385960448/STEM/6a7de21ee6424115a15666bc7d053c59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/2016/1/22/1572463379816448/1572463385960448/STEM/88b45c8aad22499b9e1d892eb8916d6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/99b16cff607cdc2d69afc70dc778acbb.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/2016/1/22/1572463379816448/1572463385960448/STEM/d0f4ba95fbef46a7b1236dc8a97b313b.png)
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2 . 如图四边形
为菱形,
为
与
交点,
平面
,
(1)证明:平面
平面
;
(2)若
,
,
,令
与平面
所成角为
,且
,求该四棱锥
的体积.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/17/4f341222-7cd8-44c5-88db-5894ab3e6bf5.jpg?resizew=212)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662698361c6b3ddaf0c28a3c87be53e0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4a46fbde58e12b1edc038ae9e921722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/926584088b939200d88e64318f2d4e6c.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e918b70b02a73685e3c536c7f380e2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7aeeee5f39ee6f9c3ea01ada75d63b93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9e06498dbb23af8854941b9ed38a582.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82f67f1e4d68dd7b2403667f7a40c69a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80c753cb1eb73fd8d136d00462970797.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/17/4f341222-7cd8-44c5-88db-5894ab3e6bf5.jpg?resizew=212)
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2卷引用:2015-2016学年湖北武汉二中高二上学期期中文科数学试卷
3 . 如图,在四棱锥P﹣ABCD中,PA⊥面ABCD,AB=BC=2,AD=CD=
,PA=
,∠ABC=120°,G为线段PC上的点.
(Ⅰ)证明:BD⊥平面PAC;
(Ⅱ)若G是PC的中点,求DG与PAC所成的角的正切值;
(Ⅲ)若G满足PC⊥面BGD,求
的值.
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735511040000/1571735516807168/STEM/0b9d263023394fb1af9c70f22b3b6dca.png)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735511040000/1571735516807168/STEM/e31015c03e16487e86bd2e8269c35cf4.png)
(Ⅰ)证明:BD⊥平面PAC;
(Ⅱ)若G是PC的中点,求DG与PAC所成的角的正切值;
(Ⅲ)若G满足PC⊥面BGD,求
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735511040000/1571735516807168/STEM/d5d15c1c1bed47e5950c1c54a24e079e.png)
![](https://img.xkw.com/dksih/QBM/2014/5/22/1571735511040000/1571735516807168/STEM/8bf0827e35bc4d63b04c955999e74e1a.png)
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6卷引用:山西省实验中学2017-2018学年高二上学期10月月考数学(理)试题
2013·上海黄浦·二模
名校
4 . 已知正四棱柱
的底面边长为2,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/17/15afdbb5-3b01-4579-b0a6-e8752930ccc3.png?resizew=135)
(1)求该四棱柱的侧面积与体积;
(2)若
为线段
的中点,求
与平面
所成角的大小.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44e898e134e65cd30d84faf0e437d3e0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/17/15afdbb5-3b01-4579-b0a6-e8752930ccc3.png?resizew=135)
(1)求该四棱柱的侧面积与体积;
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10d8eb4a9f462ca0c1d49c3fe91e720d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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5卷引用:上海市华东师范大学第二附属中学2018-2019学年高二下学期期末数学试题
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5 . 如图,在四棱锥P-ABCD中,底面ABCD是矩形,
,BC=1,
,PD=CD=2.
(I)求异面直线PA与BC所成角的正切值;
(II)证明平面PDC⊥平面ABCD;
(III)求直线PB与平面ABCD所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b8e49d68afab33806a63d25a0861c7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1720da6d65e7fa854d98322d3864240.png)
(I)求异面直线PA与BC所成角的正切值;
(II)证明平面PDC⊥平面ABCD;
(III)求直线PB与平面ABCD所成角的正弦值.
![](https://img.xkw.com/dksih/QBM/2012/7/6/1570917984845824/1570917990350848/STEM/569097308c9b467a8690e97890401a26.png?resizew=273)
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5卷引用:安徽省安庆市桐城市第八中学2020-2021学年高二上学期期初检测数学试题
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