解题方法
1 . 在正方体
中,
分别是棱
和
上异于端点的动点,将经过三点
的平面被正方体截得的图形记为
.如图中
时截面图形
为矩形.
(1)在图中作出截面图形
为梯形的情形;(直接画出图形即可,不需说明)
(2)当点
为
中点时,求
与平面
所成角的正弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1ae536809b1161fd4e83fdc7f42be96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1859959fdb4c5edd8056893f94a10a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53e97fcdcfd6183b976a61ef3222c607.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d04ea588f556c3b874b7e68ea69f49e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f805768a5ffaf8bdfa4bc3b680aafdc9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/29/5db5de67-b86d-4aaf-b4be-12f24a3ef879.png?resizew=164)
(1)在图中作出截面图形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b94469fd19f40116e2dec334919d6586.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0a851907ada2ac2c3c4880a6736d28a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ee8456443402a25b1e25d35ff7e1c98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca48c18021e7be4bbb3e95576e1c1b5f.png)
您最近一年使用:0次
解题方法
2 . 一年一度的创意设计大赛开幕了.今年小王从世界名画《永恒的记忆》中获得灵感,创作出了如图1的《垂直时光》.已知《垂直时光》是由两块半圆形钟组件和三根指针组成的,它如同一个标准的圆形钟沿着直径
折成了直二面角(其中
对应钟上数字3,
对应钟上数字9).设
的中点为
,若长度为2的时针
指向了钟上数字8,长度为3的分针
指向了钟上数字12.现在小王准备安装长度为3的秒针
(安装完秒针后,不考虑时针与分针可能产生的偏移;不考虑三根北针的粗细).
(1)若秒针
指向了钟上数字4,如图2.连接
、
,若
平面
.求半圆形钟组件的半径;
(2)若秒针
指向了钟上数字5,如图3.设四面体
的外接球球心为
,求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/26/9bc32e01-aae0-45c8-bdf6-e78ec83f4bf8.png?resizew=498)
(1)若秒针
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c023f4b501684abd869b36d6e6c7f21f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6fe3ee05321860442e84d54c6f328b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb802b0cd77d772dceff0d9ff6c879ac.png)
(2)若秒针
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7303d343202b6444bb0ccff94e48e7c.png)
您最近一年使用:0次
名校
解题方法
3 . 如图,在直角梯形
中,
,
,
,
,现将平面图形沿
折成一个直二面角,得到四棱锥
,E,F分别为侧棱
、
的中点.
![](https://img.xkw.com/dksih/QBM/2021/5/2/2712383210962944/2716480608976896/STEM/952875df-244f-44ea-9c61-a94dd71b47ad.png?resizew=132)
(1)如图,在箭头右侧画出四棱锥
的直观图(不要求精确图形);
(2)证明:平面
平面
;
(3)若
是平面
的一个法向量,求
与平面
所成锐二面角的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4864c21e9664fa9111ede6425b09563a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3185a8075eea774ea1c6298fd1d0f5af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4340375ca8abdbd6998760c944f38d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cfc3c31464d82f3eac4777b046ca69a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a45024b1f1c50249cc194e8689ec01cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://img.xkw.com/dksih/QBM/2021/5/2/2712383210962944/2716480608976896/STEM/952875df-244f-44ea-9c61-a94dd71b47ad.png?resizew=132)
(1)如图,在箭头右侧画出四棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
(2)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6501f1c913a4ef64957a2f01ab5baa15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80f747eb5b2d21c9de962cbfd4ec4bb7.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49163e9dc6878910a10d2b4aa1ddad93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b03428a8f91a5674cb8f54766c165f7e.png)
您最近一年使用:0次