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1 . 如图所示的几何体中,eqId5ce7a06ae7a34393b92b5277978ac014是菱形,eqId2807fcad45ad4b3b8490dba6af5d22e6eqIde78cce043a6c43329b55788351ff3484平面eqId5ce7a06ae7a34393b92b5277978ac014eqIdc472d67631564692992fa230585eb736eqId8e26b1a066184bf1bcbb8e885352cbfb.
说明: figure
(1)求证:平面eqId21d75cb53dbf45f6bbe6a9c4546d7179平面eqIdef6a7c356a884ffc866777aa14a0f89b
(2)求二面角eqIdb59d717e874c4c03ab64e8ecd83cbeeb的正弦值.
2 . 已知四棱锥P-ABCD的底面为直角梯形,eqId65d530aeef104876b24877e3e1ec637beqId4f36fe1415d4458893f8693fe97edb48eqIde78cce043a6c43329b55788351ff3484底面ABCD,且eqIddb2569195f7047a18e375481fe5b3bb4eqIdb60bb1649c8b41cb935771acc097167eMPB的中点.
说明: figure
(1)证明:面eqId3084aa1e379e4f168e9c2332df2ca9dfPCD
(2)求面AMC与面BMC所成二面角的正弦值.
解答题 | 较难(0.4) | 2021·北京师大附中高一期末
3 . 已知四棱锥eqIdac097205e9cb41279269aadcac3fb6f1的底面为直角梯形,eqIdab8f427008f546f2ade3d878955f299e平面eqId5ce7a06ae7a34393b92b5277978ac014,且eqId1c8d457414c04b9fa43de85ec32293d9eqId2381423d4cd146cab95f55527681a766是棱eqId0fd9ac73d66042c6bbd74f8ccdcbd9fc上的动点.
说明: figure
(1)求证:平面eqId3084aa1e379e4f168e9c2332df2ca9df平面eqId83e13ba429bb426fb39bd7720a82c64c
(2)若eqId2f8e1053799d4f20854a0c4d01574c19平面eqId0a01245d72014cd8983f1443e0d398de,求eqIdb10e81de6cca49f0a85eb66a2573a26d的值;
(3)当eqId2381423d4cd146cab95f55527681a766eqId0fd9ac73d66042c6bbd74f8ccdcbd9fc中点时,设平面eqId0170b1b3ba92400c807d09a21300c2c7与棱eqIdb53b60c33a254a27b9bfd58e818cc7e6交于点eqId517584fed25c413ba8b7bb33ffa2d5c6,求截面eqIdff1061c0574046c0882380b8aa7920f8的面积.
4 . 如图1,在直角梯形eqId5ce7a06ae7a34393b92b5277978ac014中,eqId588db7aaf8e34ddc82aa8ca1e4b643c7eqId686612a3629141cca688664f84d38e77,且eqId07e4516247794b80995d6c4ec1dc29d9,现以eqId8a76bbe21fb549e3a9c2038d58c7a3d8为一边向梯形外作正方形eqId041ea54dcbeb40308c2dd83660f2cf3b,然后沿边eqId8a76bbe21fb549e3a9c2038d58c7a3d8将正方形eqId041ea54dcbeb40308c2dd83660f2cf3b折叠,使eqIdcdb4cf40f1ef49159ac9a0676a9e0bf5eqId2381423d4cd146cab95f55527681a766eqId4eb9bc5de635473797d1f01121841811的中点,如图2.
说明: figure
(1)求证:eqId577d54bce5e84bfeabaeaeb481fefc71平面eqId272060f3416341f8924ad76b7896edd9
(2)求证:平面eqId1d7eb6bc639a4582afb23c7392e2b826平面eqId076f835a1f2c428898c79204bd1d52fd
(3)求点eqId0cd8063abf2b458f80091bc51b75a904到平面eqIdc99c444e10d84680a6fdc44586d46ce1的距离.
5 . 如图所示的多面体是由一个直平行六面体被平面eqIdfa5429391c724bd0b38b0ec703f6e802所截后得到的,其中eqId187865d0cdda4228b763296f4fb2a9c9eqIdc4ca7ed9b2a74cf482ca0e2489412f69eqId6c245634739143368cd437a12b59f3a3.
说明: figure
(1)求证:平面eqIdfc8c8677f10c4e289d5579d7cec8fad6平面eqId84c1b1e0b4d6428685b9856bbe24498a
(2)求直线eqIdd6bf126f0d644ff591fce7b3b72e5504与平面eqIdfa5429391c724bd0b38b0ec703f6e802所成角的正弦值.
6 . 如图,四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,底面eqId5ce7a06ae7a34393b92b5277978ac014为矩形,eqId790707287e604e1a97e1484c1457c054eqIdded28dbc6ca54681ab966fe6545c6cdceqIde78cce043a6c43329b55788351ff3484平面eqId5ce7a06ae7a34393b92b5277978ac014eqId93cbffaa5ae045d6ac45d1e979991c3aeqIdd293923b221045898dcc5442f88aa67a的中点.
说明: figure
(1)证明:eqId5a25e006c4334601a356a7044bd0a8c0平面eqId55a823bf7e6b4e1888e6e135a7470310
(2)若三棱锥eqIda1df1528479f4bfdbac2c1b2948e6543的体积为eqId96c7406fe9bb42d482404e87ed3f58bd,求直线eqIdb53b60c33a254a27b9bfd58e818cc7e6与平面eqIde5632d7e0a9540bfa77df0f3f1091dc8所成角的正切值;
(3)在第二问的条件下,若eqId2381423d4cd146cab95f55527681a766为线段eqId0fd9ac73d66042c6bbd74f8ccdcbd9fc中点,eqId517584fed25c413ba8b7bb33ffa2d5c6为线段eqId0627be51821d4be7b3e024a354815d64上的动点,平面eqId82bcbb2fc2d94254bc9cb84e31f7335f与平面eqId0902deb68f0b430f932caadd41cd761a是否互相垂直?如果垂直,请证明;如果不垂直,请说明理由.
7 . 如图所示,在直三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,eqIdc36c710593ee4ca18f5e8c8bb203fc22eqId23aae3fbd7eb4d2c99cac5bd0ea09b17eqIdcd415946f79047fc80d0f987dfda8ff6eqId2381423d4cd146cab95f55527681a766eqId517584fed25c413ba8b7bb33ffa2d5c6分别是eqId31581e8b21734ef88d2a14232de7e7a6eqId99a3187c2b8f4bcc9703c74c3b72f1f3的中点,给出下列结论:
说明: figure
eqId04adf724a5b94deea4ae3ea402703b93平面eqId18183224a2ed48c9a90a27df78ce56c7;②eqId756e3eb0b04b4d8f850f925f98bea289;③eqId0f359987d76046698169bf1b24016d0f平面eqId9317bc09deca4030b432cae85146dcd6;④平面eqIda665769b3d894ae981ff994e7d1e4f52平面eqIdd689ecb8ad124cf99ea52c702943dc70
其中正确结论的个数为(   )
A.1B.2C.3D.4
8 . 如图,在四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,eqId588db7aaf8e34ddc82aa8ca1e4b643c7,且eqIdd7ebb92a87bd47088420533d0415fd25
说明: figure
(1)证明:平面eqId5346c14ae189426eb85ec11c3774037f平面eqIde5632d7e0a9540bfa77df0f3f1091dc8
(2)若eqId25da5c95b1654773bdfec0ec495760b0eqIdc308a98ced0640378a1023229622524c,求四棱锥eqIdac097205e9cb41279269aadcac3fb6f1的侧面积.
9 . 如图,直三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,底面边长AB=5,BC=4,AC=3,侧棱长为eqId7bcd573cee334383ad72cc4b179fcfb3DBC中点,CEADE为垂足.
说明: figure
(1)求证:eqId710b690db05c4a9ca514d234b961f314//平面eqId578f28e5c7dc40d2b6c24754f4658494
(2)求证:平面eqId578f28e5c7dc40d2b6c24754f4658494⊥平面eqIde1eeae7b5e874850be58d303a386fde9
(3)求直线eqIddd9813abac8f48aaad0208a9f59a6bc2与平面eqIde1eeae7b5e874850be58d303a386fde9所成角的正弦值.
10 . 如图,点eqIdbedf755e0fdb4d078d6859360706b163在正方体eqId588284d93dc5489295f8f224f8e30d13的面对角线eqId3be2cc74b5df4257b116d1672859cb5e上运动,则下列结论正确的个数是(   )
说明: figure
①三棱锥eqId36d9d39d07ed46f7844c8bfdc6824842的体积不变;
eqId6113ee1e8af44d31999c240c123f4345平面eqIdef8684b13018435bb5e2d072f64022ae
③平面eqId1a42a51b5d4a4f39a86b7450ff271f51平面eqIdef8684b13018435bb5e2d072f64022ae
eqIdead44642fadc4f4096e397c394c4d8e7.
A.4B.3C.2D.1
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