1 . 已知正方体
的棱长为1,H为棱
(包含端点)上的动点,下列命题正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
A.二面角![]() ![]() |
B.![]() |
C.若O在正方形![]() ![]() ![]() |
D.若![]() ![]() ![]() ![]() |
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解题方法
2 . 如图,在正方体
中,E为棱
上的一个动点,F为棱
上的一个动点,则直线
与平面EFB所成的角可能是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/19/5782a9a4-6976-43e3-afc2-a1d0d71bc530.png?resizew=172)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22adbc0da438220f9cace11b629d799b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/19/5782a9a4-6976-43e3-afc2-a1d0d71bc530.png?resizew=172)
A.![]() | B.![]() | C.![]() | D.![]() |
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解题方法
3 . 如图,在棱长为2的正方体
中,M,N分别是
的中点,则( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/20/222d76a3-4880-449a-b491-f061077999e6.png?resizew=185)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8befa3a8f3880668c5ff01dd0e62141.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/20/222d76a3-4880-449a-b491-f061077999e6.png?resizew=185)
A.四点A,M,N,C共面 |
B.直线![]() ![]() ![]() |
C.异面直线![]() ![]() ![]() |
D.过M,B,C三点的平面截正方体所得图形面积为![]() |
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4 . 如图,在正方体
中,E、F分别是
、
的中点,G为线段BC上的动点(含端点),则下列结论中正确的是( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/18/b3e56163-4cbe-4256-b13f-d94864215bc7.png?resizew=173)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/394c5d2f55221975503be8aa18022480.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f66fb71b75b63594ebeeeebd1963eed5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/18/b3e56163-4cbe-4256-b13f-d94864215bc7.png?resizew=173)
A.存在点G使得直线![]() |
B.存在点G使得直线AB与EG所成角为45° |
C.G为BC的中点时和G、C重合时的三棱锥![]() |
D.当G与B重合时三棱锥![]() |
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解题方法
5 . 如图,在直三棱柱
中,
,
,
,点P,R分别是棱
,CB的中点,点Q为棱
上的点,且满足
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/16/eb982da7-85df-4af7-b9d2-5838d141a4f2.png?resizew=145)
(1)证明:
平面AQR;
(2)求平面PQR与平面AQR夹角的正切值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b935580f6c20b82112df78d570a482b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd7e3b114896f000678d4ff3f580e76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/259414bba09b088ca98ed4c3fcee8197.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/455cfc19d4112a3b508304f889ff74cd.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/16/eb982da7-85df-4af7-b9d2-5838d141a4f2.png?resizew=145)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c584de295efd542134b4ecb419c6ea17.png)
(2)求平面PQR与平面AQR夹角的正切值.
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6 . 如图,在棱长为2的正方体
中,E,F,G分别为AB,BC,
的中点,点P在线段
上,
平面EFG,则( )
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/18/d60ecb03-335a-4462-8207-d07605b75749.png?resizew=179)
![](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/8cfbc0b5a8fbde804bd8425a4b76d207.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/574a6cff0aca240f071ed933c28e32f6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/18/d60ecb03-335a-4462-8207-d07605b75749.png?resizew=179)
A.![]() ![]() | B.点P为线段![]() |
C.三棱锥![]() ![]() | D.平面EFG截正方体所得截面的面积为![]() |
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7 . 如图,在三棱柱
中,
平面
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/22/bf82e2bc-83f6-438d-a1d2-e319e78f62f4.png?resizew=207)
(1)求证:
;
(2)若
,直线
与平面
所成的角为
,求二面角
的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78f521b2a852e44bee4c553251d37d44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58cc6184b191e6da43911e701121517e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/22/bf82e2bc-83f6-438d-a1d2-e319e78f62f4.png?resizew=207)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7870cee007535b979d35bc7feab75616.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ee950f7e202d647d53efa55d48e01b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58cc6184b191e6da43911e701121517e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/996bf3d35b6763cbc1a423b13a9df2dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/513d1fb90e198a552a61d535187a2fe5.png)
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8 . 如图,三棱柱
的棱长均为2,
为
的中点,平面
平面
,平面
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17f1396f669e6070a07115596c6bcd3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/09185985-b7eb-4227-b77c-f98544036c38.png?resizew=191)
(1)求证:
平面
;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71f2185273bf04c11118c7954f7ec822.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/653ddc9c58281c1cc096de7c15ed0749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9170ab5963ca15e9a35075c0781dde7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d17f1396f669e6070a07115596c6bcd3.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/28/09185985-b7eb-4227-b77c-f98544036c38.png?resizew=191)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d020a9a555c8992a24992d63a4981bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/782ad39b04e9d1c7b4e774df85c53b1f.png)
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9 . 所谓正三棱锥,指的是底面为正三角形,顶点在底面上的射影为底面三角形中心的三棱锥,在正三棱锥
中,
是
的中点,且
,底面边长
,则正三棱锥
的外接球的表面积为__ ;
与底面
所成角的正弦值为__ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0efbdc4cbc82cc55bed9905f81fbfc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21ea52361458ce2e49ed0fe99d8e6c02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
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10 . 如图,四棱锥
中,
,
,
,
为正三角形.若
,且
与底面
所成角的正切值为
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/5/cfc488d9-5242-46cd-bcd7-56f6e1539ed8.png?resizew=150)
(1)证明:平面
平面
;
(2)
是线段
上一点,记
,是否存在实数
,使二面角
的余弦值为
?若存在,求出
的值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88f86b6bb8d0612e06f5579090727379.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5acb763021bf166ca719d07223591d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acee03d4bb4667b6c345221b6c9b0fa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/967f74b8993c61634ceed95edca05ffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d5989c84e320b504511f23eeb6e7357.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/5/cfc488d9-5242-46cd-bcd7-56f6e1539ed8.png?resizew=150)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4aa9084b8fe0fe05c4388d1f835587b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7587206b2da150be5e76512d13b368e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/564cfa0a1a08620d4b70b155337fbf42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/64f1145c162038df3c7184d9201c628e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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