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1 . 如图,在直四棱柱eqId588284d93dc5489295f8f224f8e30d13中,底面四边形eqId5ce7a06ae7a34393b92b5277978ac014为梯形,点eqId2efda802d5534f6d92d9f8af7aaec28beqId99a3187c2b8f4bcc9703c74c3b72f1f3上一点,且eqIdcd4b64740a464c1fa017687eb13bc142eqId588db7aaf8e34ddc82aa8ca1e4b643c7eqId7ed346a925864e5d9078c8661ee2edce
说明: figure
(1)求证:eqId30b2a53b796243f1b1681f9e13bfc74b平面eqId8297c1bd76e44378b14f6b32270a7c64
(2)求点eqId19a4eb16029e4550a14f2afe4741a3c3到平面eqId30595f3a330644eca990f7502b828b5f的距离.
2 . 《九章算术》是我国古代的数学著作,是“算经十书”中最重要的一部,它对几何学的研究比西方要早1000多年.在《九章算术》中,将底面为直角三角形,且侧棱垂直于底面的三棱柱称为堑堵.如图,在堑堵eqId9881c5faa5cb449bb37fa59b41c76e43中,eqIdfb4e495ba1384cb8895c57c4ec5236e5eqId9a43edfb78b548ea8d2ade9718631891MN分别是eqId8f39d706a505485987ccbb3f6177d72dBC的中点,点P在线段eqId31581e8b21734ef88d2a14232de7e7a6上.
说明: figure
(1)若PeqId31581e8b21734ef88d2a14232de7e7a6的中点,求证:eqIdd87757ad2a774cd5a0ac708180ef8a33平面eqId38df751166484f09b6aeaf66df4e2d17.
(2)是否存在点P,使得平面PMN与平面ABC所成的二面角为eqIdd01b7918993a412cb774dea127f210b6?若存在,试确定点P的位置;若不存在,请说明理由.
3 . 如图,已知四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,eqId43fe05c727b14343a0211870697669c9eqId5ce7a06ae7a34393b92b5277978ac014eqIdd9b3ca0a3e8a410392e4a467073b71d2eqIdf63c69754bdf4ac38e17101194073067eqId93cbffaa5ae045d6ac45d1e979991c3aeqIdb53b60c33a254a27b9bfd58e818cc7e6中点,eqId63db14a5b4334f3ea583c8fb12b0d175eqId34c6b6e427744abfb88dc90b8c7c2111上,eqIddf9c9a3e2d69491892c10119156d4938eqId43ac27a50da4401eb00f9dc163fb7161,则下列结论正确的是(   )
说明: figure
A.eqIdd1c000f77c224125baf22c8802a6d817eqIde5632d7e0a9540bfa77df0f3f1091dc8
B.eqId0fd9ac73d66042c6bbd74f8ccdcbd9fc与平面eqId5ce7a06ae7a34393b92b5277978ac014所成角为30°
C.四面体eqIdbcf41012ae0d4979a1561c82c687b8db的体积为eqIdbf4d8b310ca74c22a7eb63a406f4674a
D.平面eqId5346c14ae189426eb85ec11c3774037f平面eqIde5632d7e0a9540bfa77df0f3f1091dc8
4 . 如图,长为4,宽为2的矩形纸片eqId5ce7a06ae7a34393b92b5277978ac014中,eqId93cbffaa5ae045d6ac45d1e979991c3a为边eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,将eqId5f7198bcbf3f4e21b941b37501b323df沿直线eqId66d409c343f04cfda647f9bfc7da6b2a翻转至eqId67ad11ba5be64c5e9cb7e7a50589ac66(eqIddffedcea7d1945a391bf8d2686bd83a3平面eqId5ce7a06ae7a34393b92b5277978ac014),若eqId2381423d4cd146cab95f55527681a766为线段eqId710b690db05c4a9ca514d234b961f314的中点,则在eqId1ba8a6a849e04b31abd4a64a759061b5翻转过程中,下列正确的命题序号是___________.
说明: figure
eqIda79ed079456a411887e217d3338fadfd平面eqIde932a97fc208440ea8df9ac8c7f6baf9
②异面直线eqIda91c1cc3600e4aabb12a8ccd0ca4ce8eeqId3dd88fd134c1472886302db0246dfc44所成角是定值;
③三棱锥eqIdbb2c0eb834254e9bba71990221df485a体积的最大值是eqIdd1ebfa5c81a04d18900d19595d070677
④一定存在某个位置,使eqId86761d9342cc4ddd87c727bbc3d68c20
5 . 如图所示,在矩形eqId5ce7a06ae7a34393b92b5277978ac014中,eqId9244d0dcab2d4e348a49df61d474ae87eqIdb60bb1649c8b41cb935771acc097167e,点eqId93cbffaa5ae045d6ac45d1e979991c3a是线段eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,把三角形eqId7fe58cc7d96542c3a7f7b3689dee0c96沿eqId66d409c343f04cfda647f9bfc7da6b2a折起,设折起后点eqId052844cae8574a8ab842c38a039baac0的位置为eqIdbedf755e0fdb4d078d6859360706b163eqId63db14a5b4334f3ea583c8fb12b0d175eqIdd293923b221045898dcc5442f88aa67a的中点.
说明: figure
(1)求证:无论eqIdbedf755e0fdb4d078d6859360706b163在什么位置,都有eqIdfec789eef46d41f085a9535585488fc5平面eqId6bc768e544bb4cc683314bfcee6add55
(2)当点eqIdbedf755e0fdb4d078d6859360706b163在平面eqId5ce7a06ae7a34393b92b5277978ac014上的射影落在线段eqId66d409c343f04cfda647f9bfc7da6b2a上时,若三棱锥eqIdde815db214c14e8b994be53029c90d4f的四个顶点都在一个球上,求这个球的体积.
解答题 | 一般(0.65) | 2021·浙江高二期末
6 . 如图,在四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,eqId71244e6ba07c460fa53ad8370a5a0d0d底面eqId5ce7a06ae7a34393b92b5277978ac014,底面eqId5ce7a06ae7a34393b92b5277978ac014为梯形,eqIdeec1827a5d2841888521a9471977822deqIdf0ad309b98cf449cbb8337899f6fa4aa,且eqIda3e4dbffea7d4365bbc5ddbb8085b0a6
说明: figure
(1)若点FeqIdd293923b221045898dcc5442f88aa67a上一点且eqId70bfbc83f14c4730abda57b4c978cee8,证明:eqIde171c0f98d4c4668b6b2783bc491091e平面eqIdfcd3aba6b7804fb68706ce7697ffa7a3
(2)求直线eqIdd3a908032e2b40df95a6d86089b3f682与平面eqId8d5158cfa6f04afc87480399685f125f所成角的正弦值.
更新:2021/06/16组卷:0
7 . 如图所示,在三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,M为棱eqId17ceafc9b1a74a37a89907801af1e742的中点.
说明: figure
(1)求证∶eqId3be2cc74b5df4257b116d1672859cb5eeqIdaac131017fe1430fb2c218bad7fdccd2平面eqIddbbcdc1769324f6a9bae763d6573e39a
(2)若eqIdea82c30b37894b1bb742dee61200db07⊥平面ABCeqIdfb4e495ba1384cb8895c57c4ec5236e5AB=AC=AA1=2,求点B到平面AB1M的距离.
8 . 如图,已知四边形ABCDBCEG均为直角梯形,eqIdeec1827a5d2841888521a9471977822deqIddae5910651ce4c7585eaa0d573600e9d,且eqId0c4ce604ecc34941b669c7bac03cbfe2eqId27b24f6ed30043c0b9aa2ff01873776deqId7b3a67a005e644c680f45da81b5fd6be.
说明: figure
(1)求证:eqId2a18056773804fa4b2f9908017f1e69d平面BDE
(2)求三棱锥eqId06f0796d441c46ad9c8384230a02b091的体积.
单选题 | 较易(0.85) | 2021·江苏高一期末
9 . 若eqId70a27b6ddf6b478285353abb3b1f3741eqIdaea992e70d4943e49e893817eb885ed7是两条不同的直线,eqIdc13953f2514e4c8f9c8aaaf5241c33aceqId4eb56f42ca674f2f9c9101b548763159是两个不同的平面,则下列命题正确的是(   )
A.若eqId6553adfc2daa459d8d01567a19498914eqId342a1e9f4d8d444abd2df388b0e45f01eqIdf480d8cc26064a1fabcd202b5b2c9f6c,则eqIdeb392e4fe33541f9aaa84a41ee853034
B.若eqIdf480d8cc26064a1fabcd202b5b2c9f6ceqIde01c1c26a9b24736b83a563bffe6e2d9eqIdd02fd5ff61524385a556673ea1b32106,则eqId6553adfc2daa459d8d01567a19498914
C.若eqIdf098609c70d0451dbc7903caecd2d08eeqId1b64978a1bbd40d58cb8eadb0e26b000eqId5e33f82eb88a47c09bb62de36d7f6f23,则eqId6cbd007ab49741748d41c4c7bdb977b5
D.若eqId6cbd007ab49741748d41c4c7bdb977b5eqIde01c1c26a9b24736b83a563bffe6e2d9eqIdd02fd5ff61524385a556673ea1b32106,则eqIdd80209f32d0945f7a14a286b64d35d96
更新:2021/06/15组卷:0
解答题 | 较难(0.4) | 2021·浙江高一期末
10 . 如图,在四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,eqId7a004c9db657435db952be9a2365dae5为正三角形,底面eqId5ce7a06ae7a34393b92b5277978ac014为直角梯形,eqIdeec1827a5d2841888521a9471977822deqIdca511c083a3a41c3aad22d3fa8c11c50eqId58945b7772734107a9cd5346562089ca,点eqId71891062a8724584af9a2c4c77cd566e分别在线段eqId8a76bbe21fb549e3a9c2038d58c7a3d8eqIdb53b60c33a254a27b9bfd58e818cc7e6上,且eqId30de623543cb4be9aef08491690af687
说明: figure
(1)求证:eqIdd001fdcd675d4e069c77689f5b05a2fc平面eqId8a53af4412f542e7a1819df0a17de410
(2)设二面角eqIdab2dde256eda44e08c4cdd9d7595bf53大小为eqId6c9518e43fd5459798a30cfc10026e3c,若eqIdb639d1513764475d8103007ed9382696,求直线eqId1b51efe7c2fa42748ac5a6be262e2fa4和平面eqIde5632d7e0a9540bfa77df0f3f1091dc8所成角的正弦值.
更新:2021/06/14组卷:0
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