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1 . 如图,在棱长为4的正方体eqId588284d93dc5489295f8f224f8e30d13中,设EeqId8f39d706a505485987ccbb3f6177d72d的中点.
说明: figure
(1)求证:eqIdcbdc5370a6bb446f95bfcb9853e7b641平面eqIde34e90d025da45a6a62834f792e92274
(2)求三棱锥eqId9df060ead733413487e5b89b703101e9的体积.
2 . (多选题)如图,正方体eqId588284d93dc5489295f8f224f8e30d13的棱长为eqId70a27b6ddf6b478285353abb3b1f3741,线段eqIddf87b224fa464be9b9fb9ac301b216fb上有两个动点eqId93cbffaa5ae045d6ac45d1e979991c3aeqId63db14a5b4334f3ea583c8fb12b0d175,且eqId6315ca0a47f8468ba5ba69d26e849abd,以下结论正确的有(   )
说明: figure
A.eqId64c4c5e3990c45b5b5137f876e933707
B.点eqId052844cae8574a8ab842c38a039baac0到平面eqId3f2d5304f1fd44b1808666be2c3f28aa的距离为定值
C.三棱锥eqId18223a5cfa234877b2e7d18ccf827197的体积是正方体eqId588284d93dc5489295f8f224f8e30d13体积的eqIdd056375c2cc745268c01dc2cbaa35258
D.异面直线eqId7111d4972b7c401c903ff4927c13b5eceqId91a3a60f05ba4520a4dfd21420cf28ca所成的角为定值
填空题 | 一般(0.65) | 2020·河北武强中学高三月考
3 . 在长方体eqId588284d93dc5489295f8f224f8e30d13中,eqId309bf7da6b2748ebb01e016772c0e680eqIda59d0dabefd241628cfd45f5506922c6eqId7229fa02543f40cabe1bc0aed8524b92,若体对角线长为eqId295fc8b6b6304713ba8fa91145a7091e,则长方体的体积的最大值是__________.
4 . 如图,多面体ABCDEF中,四边形ABCD为菱形,在梯形ABEF中,eqIdc59d326049964cdb82980303ab0ace51eqIda0ea6d904beb4737a0140eae5b32c6c5eqIdf5202a838a6045d4927d18a37add5657,平面eqIdad819b6869f44adcae2321faa92686d9平面ABCD
说明: figure
(1)证明:eqId4d325781b26648999078d67577538f5d平面AFC
(2)若多面体ABCDEF的体积为eqIdf7d6b53ad41c46fdaa1cbdb9c26ced4deqId906777a8e78b4507b8aaa868ef68d96e为锐角,求eqId906777a8e78b4507b8aaa868ef68d96e的大小.
解答题 | 较易(0.85) | 2022·北京东城区·高三专题练习
5 . 已知如图1所示,等腰eqId64b9b599fdb6481e9f8d9a94c102300b中,eqIdbcc63cd6ef88410ea6e9264a87594e92eqId8d343c4ae3c848da834d48a461840a6aeqId312ef2c826304089b9b0f1d1e88b0f50eqId0627be51821d4be7b3e024a354815d64中点,现将eqId05196544fb324d98ac235aa95d6bfb2e沿折痕eqId8a76bbe21fb549e3a9c2038d58c7a3d8翻折至如图2所示位置,使得eqId037298073c0f4a818ff1141aea85c5c3eqId93cbffaa5ae045d6ac45d1e979991c3aeqId63db14a5b4334f3ea583c8fb12b0d175分别为eqId99a3187c2b8f4bcc9703c74c3b72f1f3eqIdcf2da96900c948a1b3ce7cbfd420c080的中点.
说明: figure
(1)证明:eqIdcf00ae9da63d4881af77194726b4e72a平面eqIda3eb5356d699417e8d2a360a77f14a6b
(2)求四面体eqIdda27baf162634ddaab4c30f70ed6eb60的体积.
更新:2021/09/17组卷:0
解答题 | 较易(0.85) | 2022·北京东城区·高三专题练习
6 . 如图所示,在三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43中,平面eqId4d9c1d86a7444e3bbcc0abc83923f1d8平面eqIda670baf5d73f4d7689915a00c51820f2eqId57abc732a43b4867aa4478942b19d047eqId8a46d4e84cb64ede9033a0d8ae9adef3eqId312ef2c826304089b9b0f1d1e88b0f50eqId3f5a5d0fed7d4dc1bcce73cf1b7a92d3分别为eqIdcf2da96900c948a1b3ce7cbfd420c080eqId17ceafc9b1a74a37a89907801af1e742的中点,且eqId232cafc94bef4f6494a661d5336a89de
说明: figure
(Ⅰ)在棱eqIdea82c30b37894b1bb742dee61200db07上是否存在点eqId2381423d4cd146cab95f55527681a766,使得eqId6a613909bf614492bb0f4d49f8d2cd4a平面eqId30595f3a330644eca990f7502b828b5f?若存在,请找出点eqId2381423d4cd146cab95f55527681a766的位置;若不存在,请说明理由;
(Ⅱ)求三棱锥eqId5e1e7d7742274ce7be2860afb11f6508的体积.
更新:2021/09/17组卷:0
单选题 | 容易(0.94) | 2021·黑龙江齐齐哈尔市·高一期末
7 . 中和殿是故宫外朝三大殿之一,位于紫禁城太和殿与保和殿之间,中和殿建筑的亮点是屋顶为单檐四角攒(cuán)尖顶,体现天圆地方的理念,其屋顶部分的轮廓可近似看作一个正四棱锥.已知此正四棱锥的侧棱长为eqIdc24d2a32e60648a191c59793786fb26e,侧面与底面所成的锐二面角为eqId6c9518e43fd5459798a30cfc10026e3c,这个角接近30°,若取eqId99f8947e6f18470c867deddcdef27efd,则下列结论正确的是(   )
说明: figure
A.正四棱锥的底面边长为48m
B.正四棱锥的高为4m
C.正四棱锥的体积为eqId3937aa3b2fb740339db2c24ae347e0fa
D.正四棱锥的侧面积为eqId1910fafef57442548e7bbcdc809e465c
8 . 如图,正三棱柱eqId9881c5faa5cb449bb37fa59b41c76e43的底面边长是2,侧棱长是eqIda2a0ce3781fa4c26b624f5966b7dee44eqId312ef2c826304089b9b0f1d1e88b0f50eqIdcf2da96900c948a1b3ce7cbfd420c080的中点.
说明: figure
(1)求证:eqIda5b89db3436946ed929b535481b22f5d平面eqId7f3b05bea9df4e50902fe80cbca666a1
(2)求三棱锥eqId9c307ae2a74246fea9fb1f3999b9a503的体积.
9 . 如图,在四棱锥eqIdac097205e9cb41279269aadcac3fb6f1中,平面eqId4c55545bc966406cad92a6b34878babc平面eqId5ce7a06ae7a34393b92b5277978ac014eqIdf59ce247769044b5959585d69a1bd977eqIdc1f6247143e74ee5a542daba2b5ee45feqId6e82d3b62ac24ed5b35425ee7b513530eqId66fde4909a81403084faa2802d7c7aefeqId93cbffaa5ae045d6ac45d1e979991c3aeqIdb53b60c33a254a27b9bfd58e818cc7e6的中点.
说明: figure
(1)证明:eqId1163d28b430c4f5899f89f5ee3e144e4平面eqId233ab96cfe1e48929ef4b707165655bd
(2)若eqId75449bcf06924e0592f1eb6775fce8b1,求三棱锥eqId07d7b5317e4a4bd7a31ad7a607b76ad9的体积.
10 . 棱长为2的正方体的外接球的体积是(   )
A.eqIdc7bebb2453684a71aa6e7cea322658d7B.eqId054411e869d34199a45ac00b267a0409C.eqId139ee5dbf44a4dbc8ee868550b97c9f6D.eqId82b4c1c63963467fbef8eeaf795e6131