如图所示,在直四棱柱
中,底面ABCD是等腰梯形,
,
,
,四边形
是正方形.
![](https://img.xkw.com/dksih/QBM/2022/5/24/2986420541005824/2987744622305280/STEM/0babbae5-66e1-4454-955f-aeecba309a60.png?resizew=245)
(1)指出棱
与平面
的交点E的位置(无需证明),并在图中将平面
截该四棱柱所得的截面补充完整;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e09725691ee7851f54c0dee86b2bf55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fff774b4b0087a6f304ce930d359be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12143a06ed24558d8cc7ad39961d3e1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f945a69cf7e8213e50622125cde652f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b724168afaee2ecddf97257180be18.png)
![](https://img.xkw.com/dksih/QBM/2022/5/24/2986420541005824/2987744622305280/STEM/0babbae5-66e1-4454-955f-aeecba309a60.png?resizew=245)
(1)指出棱
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d88bf46ad08f9677c37eed1d0369329.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b2213b575a7cfaffcdf91885005c7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36b2213b575a7cfaffcdf91885005c7d.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8375f8f2a0dd3e0212ce52d334952c.png)
更新时间:2022-05-26 15:24:54
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解答题-作图题
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【推荐1】请在长方体模型中找出三个角都是直角的空间四边形,并画出该图形.三个角都是直角的四边形一定是矩形吗?
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【推荐2】如图是三个空间图形的平面展开图,请问各是什么空间图形?
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解答题-证明题
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【推荐1】在空间直角坐标系
中,以坐标原点
为圆心,
为半径的球体上任意一点
,它到坐标原点
的距离
,可知以坐标原点为球心,
为半径的球体可用不等式
表示.还有很多空间图形也可以用相应的不等式或者不等式组表示,记
满足的不等式组
表示的几何体为
.
(1)当
表示的图形截
所得的截面面积为
时,求实数
的值;
(2)请运用祖暅原理求证:记
满足的不等式组
所表示的几何体
,当
时,
与
的体积相等,并求出体积的大小.(祖暅原理:“幂势既同,则积不容异”.意思是:所有等高处横截面积相等的两个同高立体,其体积也必然相等)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e336d6ca2cae3d6e6c3810d7e521a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9437ae697faa99579163106aa5a62e84.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/848c9fd2f04fb6d9f9cec5f7c8756ca8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c3ff0b0fea1cb642d3f6be77a1ff32f.png)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfcff373b650f57e068b74b3356a9f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c3ff0b0fea1cb642d3f6be77a1ff32f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb9c018281fcaaf52863e1f83d9dad0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3eabd5f3a86afe49dcd70571e2b96cfd.png)
(2)请运用祖暅原理求证:记
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b9cb8e6ff801523b0304576cd69fd2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff8b3a41fd9d00b2c99425c1f7529639.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6eaa137a2290a9a9ec7ad635d17dbb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfcff373b650f57e068b74b3356a9f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6eaa137a2290a9a9ec7ad635d17dbb6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c3ff0b0fea1cb642d3f6be77a1ff32f.png)
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解答题-问答题
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解题方法
【推荐1】四棱锥
中,底面
为直角梯形,
,
,
,
,
,
为
的中点,
为
的中点,平面
底面
.
![](https://img.xkw.com/dksih/QBM/2020/5/21/2467717167136768/2468822116278272/STEM/892fe585-732d-4a80-8b1b-6a5965cb85e4.png)
(Ⅰ)证明:平面
平面
;
(Ⅱ)若
与底面
所成的角为
,求二面角
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/d5acb763021bf166ca719d07223591d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a0e5697eca3f5205cb7b343648240bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e921f46d90e43f4517c55832b6280f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62974d34de3a12418d6b700420afd1b2.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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93edc7bb513f40a89173121c8570cd65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://img.xkw.com/dksih/QBM/2020/5/21/2467717167136768/2468822116278272/STEM/892fe585-732d-4a80-8b1b-6a5965cb85e4.png)
(Ⅰ)证明:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e51838e395dfc9d9ef597d9e01f46272.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852aabd89edffc1b94344ff3f1f31ccd.png)
(Ⅱ)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d88591679796c52024d11c4de641bdb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b051f51b7d4ce0a3a6d0b7eb880a2fbf.png)
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【推荐2】如图,以正四棱锥
的底面中心O为坐标原点建立直角坐标系
,其中
,
,E为VC的中点,正四棱锥的底面边长为
,高为h.
![](https://img.xkw.com/dksih/QBM/2021/12/4/2865417694044160/2865551842148352/STEM/17b123a8-3566-457e-ba62-dd4ea5c75904.png?resizew=273)
(1)求
;
(2)当
是二面角
的平面角时,求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ebfcf34539673d516eb9b259951a81ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5e336d6ca2cae3d6e6c3810d7e521a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7f133f25700c0c90d5cfd19ef558571.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6836b1016a339232c2e2a5266b868144.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/878e89b6eca35e34c863e832a2c661db.png)
![](https://img.xkw.com/dksih/QBM/2021/12/4/2865417694044160/2865551842148352/STEM/17b123a8-3566-457e-ba62-dd4ea5c75904.png?resizew=273)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b7676b62a098d033ae2468327c2e2f2.png)
(2)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb3de20ba8bda63c650ef571be155702.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080357c95392e1c1831d999893ec6e77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9be5bdd3397ac98037ccd216ead50c39.png)
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解答题-证明题
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【推荐3】已知平行四边形
中,
,
,
,
是线段
的中点,沿
将
翻折到
,使得平面
平面
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/23/b3d36c9a-207e-4e76-90ba-e1b5a48a6b27.png?resizew=196)
(1)求证:
平面
;
(2)求二面角
的余弦值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df6d51738ac1bc8b9530ea4a55745c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2edea101a3dbb3aa5c7669b4710d108a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66549859b7597fb32ad03d1b34fc6d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/23/b3d36c9a-207e-4e76-90ba-e1b5a48a6b27.png?resizew=196)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2621f34678502074627ec62b46ac8f6c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca67a5b8f69507c8b80379e86f90a8ce.png)
(2)求二面角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1e4450a48d31b761acb861c86879657.png)
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