1 . 如图,在四棱锥
中,
底面![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1abdd8aa7ad17cafae47c8c855c1d682.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb31ef428bd9de9bc875b343feded3c7.png)
,
.
;
(2)求平面
与平面
的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a1b49f64e0065edad868b25e9fcada3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1abdd8aa7ad17cafae47c8c855c1d682.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb31ef428bd9de9bc875b343feded3c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd830ebaa37263512054ba12546609c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f4b4c90faa3f4e43393b38400ff44b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/712f7375b4ede5f75c0d81870c0f86af.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7abd284f76d9f5769bc189508ce2572b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
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2 . 在三棱柱
中,侧面
平面
,
,侧面
为菱形,且
为
中点.
平面
;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42d3a82b8e587ee890467835bc4e854c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df00cdf77ed39ca5a0b305861a693142.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a65220a73f2926dd73e083627524af97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/269c684310d0f7b5b9bf0a291e7ee748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d9a8181f7a7fe7f3fac872ce9534f15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18a0bebcc9e80bbe35943d42f0e00d04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4890e58791814622b87c4d60ea971f54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f96c673a2381f118ea2d3efc0bca1f3.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd9ed960d92cd08e368ed56651fe6632.png)
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解题方法
3 . 如图,在四棱锥
中,
底面
,在直角梯形
中,
,
,
,
是
中点.求证:
平面
;
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0585b6c0f156eecf9662b9846d4eb693.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f1897a7e856b42f8cee0f286ad913d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1134c8e3440abb6cd385af2c169037fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae1e04eeb4de72e5750dae77bcb6f88a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fa6ea683971fa8b6299d7aab6d04092.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/932a04304f2d4975955d4baabb2deeea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e582d73b96ba649378379c3074d506d.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4cd8ba7eb52e38857830162e770f534.png)
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4 . “阳马”是我国古代数学名著《九章算术》中《商功》章节研究的一种几何体,即其底面为矩形,一条侧棱垂直于底面的四棱锥.如图,四边形
是边长为3的正方形,
,
.
是一个“阳马”;
(2)已知点
在线段
上,且
,若二面角
的余弦值为
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91d5a57d368261e7a0a61d8386459eea.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8cee96ca90f9f26644860329443ed56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/faeb97acf19bd3b2c6c77c2814df4d2f.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/69e27dea946df6947fb791374c992dcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6f351b3ffc75878acdbbe4d4926524f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5930602f8d9bb301d34db872d7a3cd53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df64046e91b047037f19e4032e3b6de3.png)
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解题方法
5 . 已知正方体
的棱长为1,点P是底面正方形
对角线
上一动点(含端点),则( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
A.![]() ![]() |
B.三棱锥![]() ![]() |
C.若![]() ![]() ![]() ![]() |
D.以![]() ![]() ![]() |
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6 . 在四面体
中,且
,点
分别是线段
,
的中点,若直线
平面
,且
截四面体
形成的截面为平面区域
,则
的面积的最大值为__________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65505f39fe3f011d3d84437100363d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6bce3d91ca23b86d8c6625f2632e437.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b61346bd4091070ba84a4046f87f365.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/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0047f659c182291c84c224df6b5e993f.png)
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7 . 已知
,
表示直线,
,
,
表示平面,则下列推理不正确的是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9168274ae1d5f93647f01b4455a8d42a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc7e6b27ad86982d6a36128682488f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f435efcc7869eec21bdba1ed81dc3f5.png)
A.![]() ![]() |
B.![]() ![]() ![]() |
C.![]() ![]() ![]() |
D.![]() ![]() ![]() |
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解题方法
8 . 正方形
的边长为6,
,
分别为线段
,
上的动点.
(1)若
是
的三等分点
,求
的值;
(2)当点
在
边上运动时,始终保持
,当点
运动到什么位置时,线段
最短.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71c330983114d20b630093055ac267db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/146c183051f41ac154a9ff4d74d3d5cc.png)
(2)当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bce726bceb02452bb4e5ed6b00fa94e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
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9 . 在菱形
中,
,
,将
沿对角线
折起,使点
到达
的位置,且二面角
为直二面角,则三棱锥
的外接球的表面积为( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e1f4f255d191786f7d330d278868c2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/951609a31601ae3944aba9684de61142.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04fd30d1c73fde67ff18e19dd26a420.png)
A.![]() | B.![]() | C.![]() | D.![]() |
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10 . 在多面体ABCDEF中,
,且
,
.
;
(2)若平面
平面
,求二面角
的余弦值;
(3)在(2)的条件下,求该多面体
的体积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e01f404965baabb8942547d68e02a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b49515bae8d21363bcade4512ab631c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc085e78fcc8bdb6a944962975ab1bab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ccd5c41c921836b50f8e18abfdc5df3.png)
(2)若平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9367449a5847eade07e69f4feddcb027.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ae628604c47725bb01e22dff5dca8e5.png)
(3)在(2)的条件下,求该多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
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