名校
解题方法
1 . 长方体
中,
.
(2)记(1)中截面为
,若
与(1)中过
点的长方体的三个表面成二面角分别为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adddcf2b210fdeda3e7795e779bd86aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e30816e31c2f392a4c975d539b458d89.png)
(2)记(1)中截面为
![](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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6a1f9f284b23e927ccffd063cb2d4ea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dbbde2388c030a896c364e62675190d.png)
您最近一年使用:0次
名校
2 . 如图,在四棱锥
中,
平面
,
,底面
为直角梯形,
,
,
,
是
的中点,点
,
分别在线段
与
上,且
,
.
平面
,求
、
的值;
(2)若
平面
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ccd4fd4b7a4d6b8ca0c5827c055a9ce7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b80ee363635d73f601654339028daec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce0d7095ddd69d6ceaf1065b1bc2c79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb466d3d90e7d27f27c60c8beecd4444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9154a06545db6cc85f99a1f8bb95715a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06531ed4a8fc81c1134dbe1e587b19b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79b28a07491270be75a3697538bec706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1100379a4385b9ce064847bc21760adc.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8014e499e7852b587b3b36af14b7816.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7b7c83470489253394bd288d7c920df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1afb187132164456d14a203e94cfa385.png)
您最近一年使用:0次
解题方法
3 . 在三棱锥
中,
平面
,
是
上一点,且
,连接
与
,
为
中点.
点的平面平行于平面
且与
交于点
,求
;
(2)若平面
平面
,且
,求点
到平面
的距离.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e56fdf217165748fafe938b64fa08179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c1848a0579202dc81aa65609ad60a6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4eb7e9ad5486cf1c5e506b20c5469e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50b0b858bc8fe8b2737b2febc1b3ce56.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/342d452a7b850cd3a15b23619ad39bd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56c83e3ba152215dcda525eebab11e70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6760f565c694d1cdb6d7068e14526d00.png)
您最近一年使用:0次
解题方法
4 . 如图,在棱长均为2的四棱柱
中,点
是
的中点,
交平面
于点
.
为线段
的中点;
(2)再从条件①、条件②、条件③这三个条件中选择两个作为已知,使得四棱柱
存在且唯一确定.
(i)求二面角
的余弦值;
(ii)求点
到平面
的距离.
条件①:
平面
;
条件②:四边形
是正方形;
条件③:平面
平面
.
注:如果选择的条件不符合要求,则第2问得0分;如果选择多组符合要求的条件分别解答,按第一个解答计分.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2331bccb6ebf5b9fd639df994f575a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)再从条件①、条件②、条件③这三个条件中选择两个作为已知,使得四棱柱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
(i)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11a150b9250addf8b5dbbf8a89c61c5d.png)
(ii)求点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70e11d3af986880db2910a92e26e0b5b.png)
条件①:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ddbb0422a136f45653c8c369f2d75fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件②:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
条件③:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a45953045e613b97eeee15ac188ae2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c52091eb745de866044477641a7c55f.png)
注:如果选择的条件不符合要求,则第2问得0分;如果选择多组符合要求的条件分别解答,按第一个解答计分.
您最近一年使用:0次
名校
解题方法
5 . 在如图所示的多面体中,
平面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe21d51a66caafa14054a41c9a37d1c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/836b56dbc08431a5b102a49dade806c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd4b8eb77297ee04a78626433a90b58b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b812363d7a76cc17df075a874d851ee3.png)
上求作点
使
平面
请写出作法并说明理由;
(2)求三棱锥
的高.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8d2d217e9bcd059908f117dfc4d4259.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe21d51a66caafa14054a41c9a37d1c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/836b56dbc08431a5b102a49dade806c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd4b8eb77297ee04a78626433a90b58b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b812363d7a76cc17df075a874d851ee3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8fbc229c957487495bb8cda1d4cfd8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/28f79db7c270b6ff9fb0a538ee201cfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ed46dc5ff6947bffc737c001fd1f11a.png)
(2)求三棱锥
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f178906e90bafd73e0ef9f89814855d5.png)
您最近一年使用:0次
2024·全国·模拟预测
解题方法
6 . 如图,在四棱锥
中,
,
,
,
,
平面
,点
是棱
上一点,且
平面
,三棱锥
的体积为
.
到平面
的距离;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5164a3cc47e266446d49127e2ef10c37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ad8d16722f5b9e7fd2602f14d5ffbe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/931f54178ee3eba80efdf1656b5a6fe2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00ec435aa1401dbce7863b531bf2f3e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddbb52f9b226b1db3f6f9f055948bd38.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca5dd496ee0c1170ef6dcc48266ee444.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f369bec2d5682bf6b8b317a08aff546.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9a32bd7a1b78b5a0ec562c4025aea8c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5452197f58d5f1b8e377d0f79069fbc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/802e162b98c280720fcb909cf392fda3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f9c7940d6f1ede201fd27d5d9e0650d.png)
您最近一年使用:0次
7 . 如图,在正方体
中,
,点E在棱
上,且
.
的体积;
(2)在线段
上是否存在点F,使得
平面
?若存在,求
的值;若不存在,请说明理由.
(3)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1364213f546b37f8764ddcb59e36ae4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f4b2b4c0c6650ae7e8fa57465848553.png)
(2)在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f7ba05c54b3de1f4378f7c8eb58328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9d32e76582bf550593fdef53e081225.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a98287a302228ece1fa53c5c66c590f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd1125973328ed42da7a53b457d587e3.png)
(3)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0397fff96574dbb83280ecb5fed6398d.png)
您最近一年使用:0次
2024高三·全国·专题练习
解题方法
8 . 过正三棱柱
一边
作截面,截面与底面成
,试导出截面形状与三棱柱底面边长
及高
之间的制约关系,并求其截面面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
您最近一年使用:0次
2024高一·江苏·专题练习
解题方法
9 . 如图所示,两条异面直线
与两平行平面α,β分别交于点B,A和D,C,点M,N分别是
的中点,求证:
平面α.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790fb3f95e7167bd6de71df11331f958.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5c4cd264c97c1f261229925cc5a6761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/edcf19a7f0dd0cdf59516ae585025110.png)
您最近一年使用:0次
2024高三·全国·专题练习
解题方法
10 . 如图,已知正方体
的棱长为1,
分别是线段
上靠近
的三等分点.过点
作该正方体的截面,试求截面图形的周长和面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3da8c338342e38c9aa3f274c053fd5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d689fcd45b3d2fb7627ce2051b628616.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b1bb7cb7f269ff782cc9dd01fba4e4d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a9c254087893591862787406e45c3da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee031bc1083a2d3c3a350f93503919c4.png)
您最近一年使用:0次