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解题方法
1 . 平面向量是数学中一个非常重要的概念,它具有广泛的工具性,平面向量的引入与运用,大大拓展了数学分析和几何学的领域,使得许多问题的求解和理解更加简单和直观,在实际应用中,平面向量在工程、物理学、计算机图形等各个领域都有广泛的应用,平面向量可以方便地描述几何问题,进行代数运算,描述几何变换,表述物体的运动和速度等,因此熟练掌握平面向量的性质与运用,对于提高数学和物理学的理解和能力,具有非常重要的意义,平面向量
的大小可以由模来刻画,其方向可以由以
轴的非负半轴为始边,
所在射线为终边的角
来刻画.设
,则
.另外,将向量
绕点
按逆时针方向旋转
角后得到向量
.如果将
的坐标写成
(其中
,那么
.根据以上材料,回答下面问题:
,求向量
的坐标;
(2)用向量法证明余弦定理;
(3)如图,点
和
分别为等腰直角
和等腰直角
的直角顶点,连接DE,求DE的中点坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91174b2336306191ba275a87864172b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91174b2336306191ba275a87864172b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4293abac93e7633dc4c0fef321347e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8a3b1b11c77ceb7ece55f76d2cd4618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/873c064546108a5bce78bb71bc1e4a1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea99a712a0891faf366d4fec4dde5869.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abcb5d89b04570ceda2c29e11cb27a57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/941b0d76d7b3108df49af338c989dc4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2e32257bac4199820ccae5e7bd8377cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0849dbfc3775627925de0fe2e89c1692.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb50427d2e8a7c605bbd18ea8e0c3b79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/af5f1b06a56fc382feed28e01f1ad102.png)
(2)用向量法证明余弦定理;
(3)如图,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cd99c5000629d7f49499d666e68f40d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/852b303689c31189cd47bb4a3220f9fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbfa1a2af7e38d33634c462300df381f.png)
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3卷引用:安徽省芜湖市第一中学2023-2024学年高一下学期期中考试数学试卷
安徽省芜湖市第一中学2023-2024学年高一下学期期中考试数学试卷(已下线)高一下学期期末模拟卷(范围:必修第二册全册)-同步题型分类归纳讲与练(人教A版2019必修第二册)湖南省永州市部分学校2023-2024学年高一下学期6月质量检测卷数学试题
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2 . 已知在任意一个三角形的三条边上分别向外做出三个等边三角形,则这三个等边三角形的中心也构成一个等边三角形;我们称由这三个等边三角形中心构成的三角形为其外拿破仑三角形.在锐角
中,角
所对的边分别为
且
,以
的边
分别向外作的三个等边三角形的中心分别记为
,且
的面积为
,记
为
的外接圆半径.
,求
;
(2)若
,求
面积的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38335830b93ac4d99c28a8e209eecb3f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9f054a8f158340bc788136124632325.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39900a2d790732077ffb571427a134fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b372e638a9dbed3bbf92454d816d212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a5a1f585c19f582edbb8cf73c58c6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4aa0df7f1e45f9de29e802c7f19a4f64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c0f2a532c86e3f4a1cd508839363b05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d46be1fb12ca82fdef75149d22bc1c94.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8843442efe0889ebbd393484ff8596cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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3 . 如下左图,矩形
中,
,
,
.过顶点
作对角线
的垂线,交对角线
于点
,交边
于点
,现将
沿
翻折,形成四面体
,如下右图.
外接球的体积;
(2)求证:平面
平面
;
(3)若点
为棱
的中点,请判断在将
沿
翻折过程中,直线
能否平行于面
.若能请求出此时的二面角
的大小;若不能,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4864c21e9664fa9111ede6425b09563a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4864c21e9664fa9111ede6425b09563a.png)
(2)求证:平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f04c222223dae9ef27d4c132534d9848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adaed034e575b208bdb8dca7bad66957.png)
(3)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4505508b3e36db64a207dcdaf8eb22dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adaed034e575b208bdb8dca7bad66957.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02a7ba7cd0c654714c967a900513ba16.png)
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2卷引用:安徽省级示范高中培优联盟2023-2024学年高一下学期春季联赛数学试题
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4 . 定义1:对于一个数集
,定义一种运算
,对任意
都有
,则称集合
关于运算
是封闭的(例如:自然数集
对于加法运算是封闭的).
定义2:对于一个数集
,若存在一个元素
,使得任意
,满足
,则称
为集合
中的零元,若存在一个元素
,使得任意
,满足
,则称
为集合
中的单位元(例如:0和1分别为自然数集
中的零元和单位元).
定义3:对于一个数集
,如果满足下列关系:
①有零元和单位元;
②关于加、减、乘、除(除数不为0)四种运算都是封闭的;
③对于乘法和加法都满足交换律和结合律,且满足乘法对加法的分配律,则称这个数集
是一个数域.
(1)指出常用数集
中,那些数集可以构成数域(不需要证明);
(2)已知集合
,证明:集合
关于乘法运算是封闭的;
(3)已知集合
,证明:集合
是一个数域.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57e35231b964e293122c4383dac2431b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33bfad095b65beb6e77aee3664b0e6bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e16415b61722f9961e412386e6819f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad135b14c9dcd83eab6618d7694c7b0.png)
定义2:对于一个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cc020b0997a2f37b214718112b79d8e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac385ec112e6d61b90d953e3f106ee85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29df165b5fc74dcbc39df74d541148da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb9df3a17aa370eba2add2c13cfc2619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac385ec112e6d61b90d953e3f106ee85.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c4e0c03c5aef711627f1b3124d8b5fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad135b14c9dcd83eab6618d7694c7b0.png)
定义3:对于一个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
①有零元和单位元;
②关于加、减、乘、除(除数不为0)四种运算都是封闭的;
③对于乘法和加法都满足交换律和结合律,且满足乘法对加法的分配律,则称这个数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(1)指出常用数集
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39021e0f386119bd1b5c73b843106e55.png)
(2)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37b491b231d0ff8a387813b5686579f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)已知集合
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da4c1c450038724991d6d39ecefae405.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
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解题方法
5 . 如图,将边长为2的正六边形
沿对角线
折起,记二面角
的大小为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
,连接
,
构成多面体
.
平面
;
(2)问当
为何值时,直线
到平面
的距离等于
?
(3)在(2)的条件下,求多面体
的表面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9165d9bfbb0f0d19eb482c2a4c1b29b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6238a6fb52a9d2e3521ba66ef9a5c247.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8560ca9023cf64637ce1467f338556bd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fca9eb9126c7053574c62b897582ad49.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94270844f197d524bf1da4f1385befd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3a079cfdcca9acdacecbf08f9f78cc.png)
(2)问当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3a079cfdcca9acdacecbf08f9f78cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/860884c0017c8bceb5b0edff796c144f.png)
(3)在(2)的条件下,求多面体
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fca9eb9126c7053574c62b897582ad49.png)
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2024-06-08更新
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175次组卷
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2卷引用:安徽省金榜教育2023-2024学年高一下学期5月阶段性大联考数学试题
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6 . 在平面直角坐标系
中,抛物线
经过点
和
,且在
轴上截得的线段长为
.
(1)求抛物线
的解析式;
(2)已知点
在抛物线
上,且在其对称轴右侧,点
在抛物线
的对称轴上,若
是以
为斜边的等腰直角三角形,求点
的坐标;
(3)将抛物线
向左平行移动3个单位得到抛物线
,直线
与
交于
两点,直线
与
交于
两点,若
分别为线段
和线段
的中点,连
,求证:直线
过定点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41c7a611e99d7785f66cd22fd8ee5e41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fec40ff4479edca2ed18b6cadb8db72f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/01e8b766829220981993a9bd024e6b79.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41322821ce31416fdac8dd6e0aa41c71.png)
(1)求抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
(3)将抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa841ce5d58b64f747a3c1b69bb20a5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad056c25c0fdcbcc765eb5cbc6093f2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5f5921423b5d7a0723968a8e32d974a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c0f067a2a348ceb24a408f82992eab8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7789a500686c7a73770404ead6af0590.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
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名校
7 . 如图,圆柱
的高为1,底面半径长为2,它的一个轴截面为
,点
为底面圆
的圆周上一点,且
.
是底面圆
的直径
上靠近
的一个四等分点,若经过点
在底面圆
上作一条直线与CE垂直且与圆
交于M、N两点,求线段MN的长;
(2)求平面
与平面ACB的夹角.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192f4f9446c954a291f779d963f90257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e9edc50f7febbc2d5d8dcdc23a3630a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bb5b12692517a39c320f99a479eb055.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/11ddc92d84d188c66b435664a7e7b5a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23f919bd3dde10dbbc076f7ec5149699.png)
(2)求平面
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31dba91f88e6404c86a48df67fdb6d77.png)
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名校
8 . 如图,等边
内有一动点
是等边三角形(点
在直线
两侧),直线
与直线
交于点
.
的大小是否为定值?若是定值,求出其大小;若不是定值,请说明理由.
(2)若
,求线段
长的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c856f329068123c34ad5c5c2c3dd5ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9521effc99744da7d1445af9681a3ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ed24d7acba0b3a7ebf4d520296424a3.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e72613c053b42aff06d307f2bf662ae.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
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9 . 如图,点
是正方形
的边
上一动点(异于
),连
,以
为对角线作正方形
与
交于点
,连
.
三点共线;
(2)若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/526768fab505a5637e515731c9940380.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9c15e609c856cb7856d6101aa198fef.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5ab52edccf02d432e0aaa4542a147bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/956c4a03127f33e7fb7acc3476f717ce.png)
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解题方法
10 . 已知函数
.
(1)若
,
,设函数
,请求出
的值域并求证:
;
(2)若
,
,
,记
,且
是一个三角形的三条边长,请写出方程
的所有正整数解的集合;
(3)若
是一个等腰钝角三角形的三条边长且
为最长边,求证:
在
时恒成立.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b5905520c2d7ba5536552341573fa37.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca27cc54ca0332245f5167488daa3408.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4580cc037c0c760c728cdbb74a8154c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/954e74ff18fc27295263b862e7b559fc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6a917e05cfca420bd81408cc7a02133.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e399dbac2fed2f3f99ef9cfce9b5123a.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b384412acba251d87902ab928902f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/75d508536d0c182db3e7f81a919793de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6996c86f28de1714e1ccd1c4f77aaa51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b93521270f25a0bcf1618b39808369f6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0cb6f261914d5f3fdf29325d812af540.png)
(3)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/071a7e733d466949ac935b4b8ee8d183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a71baf6217604517fd98fa97d0f55b43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4c77befb23ddbca57b9c341f5b9412e.png)
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