解题方法
1 . 已知空间直角坐标系中的四个点
分别为线段
上的动点,则下列结论正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81033af8f3c067edb20337e0f55c6008.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
A.三棱锥![]() ![]() | B.三棱锥![]() ![]() |
C.![]() ![]() | D.![]() ![]() |
您最近一年使用:0次
2 . 如图所示的几何体是由等高的直三棱柱和半个圆柱组合而成,
为半个圆柱上底面的直径,
,
,点
,
分别为
,
的中点,点
为
的中点.
![](https://img.xkw.com/dksih/QBM/2024/5/31/3508518211018752/3520553871540224/STEM/24a32289908a4049b65c769f665382cb.png?resizew=222)
(1)证明:平面
平面
;
(2)若
是线段
上一个动点,当
时,求直线
与平面
所成角的正弦值的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/209acf15985d1ea1ad86fc4a37e38c0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd8d7a1d7df296227f18a816805bf868.png)
![](https://img.xkw.com/dksih/QBM/2024/5/31/3508518211018752/3520553871540224/STEM/24a32289908a4049b65c769f665382cb.png?resizew=222)
(1)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9fc224dae99dd34e997eaaa1973edb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bff3ccea5989c60e51e321af3f53f54.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/456175ea34492f0bc025aaab668fa659.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/462b1c65b1b233ab98a90c164c0968c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d0ff310aabd2282b539537ebed3f788.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
您最近一年使用:0次
解题方法
3 . 如图,正方体
的棱长为3,E,F分别为棱
上的点,且
,平面AEF与棱
交于点G,若点P为正方体内部(含边界)的点,满足
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31be3b7305d6c181420ea7b28c420851.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f91c39222a658ebdd3da3c1ea93a17b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a25f60a2bdeb279f7742e6408831eb9.png)
A.点P的轨迹为四边形AEGF及其内部 |
B.当![]() ![]() |
C.当![]() ![]() |
D.当![]() ![]() |
您最近一年使用:0次
4 . 如图,三棱柱
中,
为正三角形,
,
,
为
的中点,
.
平面
;
(2)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92105835f8075cb75dff244e908370b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b38f9a9c04f88f40e0efccc0a3ebe6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed37556e3524d342bf19c961fde0a26c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01e5981445b6f2a6c58974158d96a4de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/982a72de174de5de98aa58b4c7d5a886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c05f5452f7682e52db629a28becb116.png)
您最近一年使用:0次
5 . 如图,在三棱锥
中,已知
,
,
,
,
,
.
为
的中点,求证:
;
(2)求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41e5db1d2fd912f77923e4c120a7dc19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080ca48cd27d4bf9d9ef084b558fc17a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ce64044898d0460aac161d93a455304.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba8c340f8945083d2993ff28eabfcccf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4d6b27a35a18492e60b5b99789d3395.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c177e06cc3f703e8ca7be7c491fa2942.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cc6f6dfdbe7d39891c35f67e1a95c7f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21665d21bbfb04410c78345de1fd15ae.png)
您最近一年使用:0次
6 . 如图所示,多面体
,底面
是正方形,点
为底面的中心,点
为
的中点,侧面
与
是全等的等腰梯形,
,其余棱长均为2.
平面
;
(2)若点
在棱
上,直线
与平面
所成角的正弦值为
,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ecc1cb55a57dde481f8dd07ab150676.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b312dab930cbbb9a4bb1a99f044dab73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/613dedf2d90e58591b7ac4a250ac7b5d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab8a2ca644d9d7cdb4784a4fd28d3904.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3653ada76ba0c8afe9d57c8e7832c6ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44fc60c0b360e1b0708a249e4ce0643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f541f7ae7c39082d202efd28805c54e.png)
您最近一年使用:0次
7 . 如图,在四棱锥
中,底面
是正方形,
,
,点
为线段
的中点,点
在线段
上.
,求证:
;
(2)若
是
上靠近点
的三等分点,求平面
与平面
所成的锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/589b38e073fdf2444c803a23a6114d9b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb8c91e4c85a9da7f54b2237d870a50d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e7e555c1ded79dfa64970c86c625b9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/316c97c5b6f7c0fbdf7b9e4b9fccb661.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/134ef0b1a2669a09f05bd4dc2496f706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fd0176c03798219fa92b44db5344086.png)
您最近一年使用:0次
解题方法
8 . 如图,在四面体
中,
平面
是
中点,
,点
在线段
上,且
.
平面
,求
的值;
(2)若
是正三角形,
,且
,求平面
与平面
夹角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63397cda22cb1fad59cf966dfb588643.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6073645d6b32ffd02450369e203ade0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e19b8bf2b4fe3a2ef4b0b08ec5f77a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10f55e0dfffeac36dd64e41fec02fb8a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57f9d682e5d3cc8573574d8d11636758.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82465b63174087aeba7788ed984583d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eef5cd98b966be5eaba09cabfe4e4ca3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
您最近一年使用:0次
解题方法
9 . 已知正方体
的棱长为2,点M,N分别为棱
的中点,点
为四边形
(含边界)内一动点,且
,则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b53e1d6a86a36dd76b41004af795d901.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/632f2bf1cd0435041fa04b01901d1c8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a7967ac1132d836e6a2ca6af5790521.png)
A.![]() ![]() |
B.点![]() ![]() |
C.存在点![]() ![]() ![]() |
D.点![]() ![]() ![]() |
您最近一年使用:0次
名校
10 . 如图,在四棱柱
中,侧棱
平面
,
,
,
,
,E为棱
的中点,M为棱
的中点.
;
(2)求异面直线
与
所成角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ecf072589c0f901d92f6bda111d841.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f79863ffcfa63117ca6741b20a48e69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/633bf2de732ae51fc06ef3d559915da0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4339a40ae9d1947ec3a4b3e2fa3a16cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2777840758e70e7dbbc18cef8f3d6d2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/594bbb40a3dddaaf9ca6bb4c222b99e1.png)
(2)求异面直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
您最近一年使用:0次