阅读与思考
下面是一篇数学小论文,请仔细阅读并完成相应的任务.
“三点共线模型”及其应用
背景知识:通过初中学习,我们掌握了基本事实:两点之间线段最短.根据这个事实,我们证明了:三角形两边的和大于第三边.根据不等式的性质得出了:三角形两边的差小于第三边.
知识拓展:如图,在同一平面内,已知点
和
为定点,点
为动点,且
为定长(令
),可得线段
的长度为定值.我们探究
和两条定长线段
,
的数量关系及其最大值和最小值:当动点
不在直线
上时,如图
,由背景知识,可得结论
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/7/138c0eff-3d7a-4bc1-aaa4-7b1e5c64e431.png?resizew=519)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/7/a8bcd1d9-87fb-401f-a910-4adb5a446c62.png?resizew=123)
当动点
在直线
上时,出现图
和图
两种情况.在图
中,线段
取最小值为
;在图
中,线段
取最大值为
.
模型建立:在同一平面内,点
和
为定点,点
为动点,且
,
为定长(
),则有结论
≥
,
.当且仅当点
运动至
,
,
三点共线时等成立.
我们称上述模型为“三点共线模型”,运用这个模型可以巧妙地解决一些最值问题.
任务:
(1)上面小论文中的知识拓展部分.主要运用的数学思想有 ;(填选项)
A.方程思想 B.统计思想 C.分类讨论 D.函数思想
(2)已知线段
,点
为任意一点,那么线段
和
的长度的和的最小是
;
(3)已知
的直径为
,点
为
上一点,点
为平面内任意一点,且
,则
的最大值是
;
(4)如图4,
,矩形
的顶点
、
分别在边
、
上,当
在
边上运动时,
随之在
上运动,矩形
的形状保持不变.其中
,
.运动过程中,求点
到点
的最大距离.
下面是一篇数学小论文,请仔细阅读并完成相应的任务.
“三点共线模型”及其应用
背景知识:通过初中学习,我们掌握了基本事实:两点之间线段最短.根据这个事实,我们证明了:三角形两边的和大于第三边.根据不等式的性质得出了:三角形两边的差小于第三边.
知识拓展:如图,在同一平面内,已知点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7191f09f447b5929962f75f52d2f4fbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d8b3ab2340d5408301b3b21dac36a31.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85533eb844a5571ccee02328a17b24e4.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/7/138c0eff-3d7a-4bc1-aaa4-7b1e5c64e431.png?resizew=519)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/7/a8bcd1d9-87fb-401f-a910-4adb5a446c62.png?resizew=123)
当动点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b65912784543a113b1e1c86e162b4537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe6926cf539cca1f818f391aa52eeb44.png)
模型建立:在同一平面内,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7191f09f447b5929962f75f52d2f4fbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fe6926cf539cca1f818f391aa52eeb44.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e8943b0c1af0a13745e715989902468.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
我们称上述模型为“三点共线模型”,运用这个模型可以巧妙地解决一些最值问题.
任务:
(1)上面小论文中的知识拓展部分.主要运用的数学思想有 ;(填选项)
A.方程思想 B.统计思想 C.分类讨论 D.函数思想
(2)已知线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04677de22b01b81ddc25b91a2cb449e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f015ed8e497b4394053ddd19683a98f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3a3a3d168a18401e47576f70a692e87.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe3a78950f6a4bb479c7ec0d8e57b5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72026e70f45cea6a7c49881592e73a02.png)
(3)已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9ccea461315a9d05aa0193b937d4bfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5992036c739557714bc05e08a93bdd3c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9ccea461315a9d05aa0193b937d4bfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dc6f90f41388dff525161c86e01f3ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe3a78950f6a4bb479c7ec0d8e57b5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72026e70f45cea6a7c49881592e73a02.png)
(4)如图4,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3e9818b013c0d75b09926cf4c6e846.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb3f9a5da641be35117fd35ba07a6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a0397c811fe80c80ecd5b871201987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/075bc6df108c9c2b3b5ac46bca7c8d3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/075bc6df108c9c2b3b5ac46bca7c8d3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37a0397c811fe80c80ecd5b871201987.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cb3f9a5da641be35117fd35ba07a6aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e7fa3aea72ccc36948a4a90f7368f71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8b5e86e15058757b1eb106d1e9faa50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
2023·山西大同·二模 查看更多[4]
更新时间:2023-05-04 12:15:55
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相似题推荐
解答题-证明题
|
较难
(0.4)
【推荐1】问题情境:如图1,P是
外的一点,直线
分别交
于点A,B,则
是点P到
上的点的最短距离.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/16/78936901-cf70-4469-a909-a484bc6e34f6.png?resizew=474)
(1)探究证明:如图2,在
上任取一点C(不与点A,B重合),连接
,
.求证:
.
(2)直接应用:如图3,在
中,
,以
为直径的半圆交
于D,P是弧
上的一个动点,连接
,则
的最小值是_______.
(3)构造运用:如图4,在边长为4的菱形
中,
,M是
边的中点,N是
边上一动点,将
沿
所在的直线翻折得到
,连接
,请求出
长度的最小值.
(4)综合应用:如图5,平面直角坐标系中,分别以点
为圆心,以1,2为半径作
,
,M,N分别是
,
上的动点,P为x轴上的动点,直接写出
的最小值为_____.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef49a3ca580a144cc65a609c167facc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/16/78936901-cf70-4469-a909-a484bc6e34f6.png?resizew=474)
(1)探究证明:如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e4b9de86f868f98ed4b982e1e2a6784.png)
(2)直接应用:如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/318cc1f1d5f09e4a2cae8059b9827ced.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
(3)构造运用:如图4,在边长为4的菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b8b98b2f83279a49e94d9f48c5e6f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/98013a5042685a1db94249e70c62c09a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8e7040c2fd8a163d71e35805775feb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e663220a66eff19da6a71e46b397db2e.png)
(4)综合应用:如图5,平面直角坐标系中,分别以点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5cbae385db0f4822d516221ffef926c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c20d0b44025a639ce3a92d639dae587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6f6558fef858bf27e9811c2d9426fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c20d0b44025a639ce3a92d639dae587.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6f6558fef858bf27e9811c2d9426fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2bb1548ddc0e5536a35b1bd78c4e7cd.png)
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(0.4)
名校
【推荐2】【项目式学习】
【项目主题】合理规划,绿色家园
【项目背景】某小区有4栋住宅楼:
栋,
栋,
栋,
栋,
处为小区入口.为方便小区居民传递爱心,物业管理处准备在小区的一条主干道
上增设一个“爱心衣物回收箱”(如图1),现需设计“爱心衣物回收箱”的具体位置,使得它到4栋住宅楼的距离之和最短.某数学兴趣小组成员开展了如下探究活动.
小组成员借助无人机航测技术绘制了小区平面图(如图2),并测量出了某些道路的长度(如表格所示),进一步抽象成几何图形(如图3),其中主干道
与
交于点
,
.小组成员又借助电子角度仪测得
.
根据图3及表格中的相关数据,请完成下列计算:
(1)求道路
的长;
(2)道路
__________米;
(3)任务三方案设计
①根据以上探究,请你在主干道
上画出“爱心衣物回收箱”的具体位置(用点
表示),并画出需要增设的小路
;
②“爱心衣物回收箱”到4栋住宅楼的距离之和的最小值为_______米.(保留根号)
【项目主题】合理规划,绿色家园
【项目背景】某小区有4栋住宅楼:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
小组成员借助无人机航测技术绘制了小区平面图(如图2),并测量出了某些道路的长度(如表格所示),进一步抽象成几何图形(如图3),其中主干道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b41f2f95d643629321deb6e905c4f1ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dc5ce02603d42d5ed598a85cfa4c3ca.png)
道路 | 长度(米) |
![]() | 40 |
![]() | 30 |
![]() | 30 |
![]() | 18 |
![]() | 32 |
![]() | 25 |
根据图3及表格中的相关数据,请完成下列计算:
(1)求道路
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(2)道路
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae248960d8c1677cf948f8251275e863.png)
(3)任务三方案设计
①根据以上探究,请你在主干道
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7daeca1219535de017a3b4f518837419.png)
②“爱心衣物回收箱”到4栋住宅楼的距离之和的最小值为_______米.(保留根号)
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解答题-作图题
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【推荐3】【问题提出】(1)如图1,在直线
上找一点P,使点P到C、D的距离之和最小;
【问题探究】(2)如图2,已知点D是
边
上一点,
.求
的长;
【问题解决】(3)如图3,在一块边长为40米的正方形
的花园中,点P是内部一点,为了有好的欣赏效果,设计者在
之间修三条小路(宽度不计),将花园分为三部分种植不同的花卉.根据调查可知修路
每米200元,修路
每米100元,修路
每米100元.测得
长为20米.设计者想知道修三条小路的费用是否有最小值,若有,若没有,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
【问题探究】(2)如图2,已知点D是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/094b80cbcc5bfa6ac24e57877e0926c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
【问题解决】(3)如图3,在一块边长为40米的正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0362386027b9a52e7f1df6bae85cca81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
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解题方法
【推荐1】阅读与理解:
折纸,常常能为证明一个命题提供思路和方法.例如,在△ABC中,AB>AC(如图),怎样证明∠C>∠B呢?
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/5/1c05d862-c93e-470d-81cc-1c096c20496d.png?resizew=405)
分析:把AC沿∠A的角平分线AD翻折,因为AB>AC,所以点C落在AB上的点C'处,即AC=AC',据以上操作,易证明△ACD≌△AC'D,所以∠AC'D=∠C,又因为∠AC'D>∠B,所以∠C>∠B.
感悟与应用:
(1)如图(a),在△ABC中,∠ACB=90°,∠B=30°,CD平分∠ACB,试判断AC和AD、BC之间的数量关系,并说明理由;
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/5/7c166599-e0ef-4d33-85e6-8a87c86f3b15.png?resizew=394)
(2)如图(b),在四边形ABCD中,AC平分∠BAD,AC=16,AD=8,DC=BC=12,
①求证:∠B+∠D=180°;
②求AB的长.
折纸,常常能为证明一个命题提供思路和方法.例如,在△ABC中,AB>AC(如图),怎样证明∠C>∠B呢?
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/5/1c05d862-c93e-470d-81cc-1c096c20496d.png?resizew=405)
分析:把AC沿∠A的角平分线AD翻折,因为AB>AC,所以点C落在AB上的点C'处,即AC=AC',据以上操作,易证明△ACD≌△AC'D,所以∠AC'D=∠C,又因为∠AC'D>∠B,所以∠C>∠B.
感悟与应用:
(1)如图(a),在△ABC中,∠ACB=90°,∠B=30°,CD平分∠ACB,试判断AC和AD、BC之间的数量关系,并说明理由;
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/5/7c166599-e0ef-4d33-85e6-8a87c86f3b15.png?resizew=394)
(2)如图(b),在四边形ABCD中,AC平分∠BAD,AC=16,AD=8,DC=BC=12,
①求证:∠B+∠D=180°;
②求AB的长.
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【推荐2】在
中,
,
,给出如下定义:作直线l分别交
、
边于点M、N,点A关于直线l的对称点为
,则称
为等腰直角
关于直线l的“直角对称点”.(点M可与点B重合,点N可与点C重合)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/1/b2c8b870-e9fc-4241-a5b4-42d28e72ffbf.png?resizew=350)
(1)在平面直角坐标系
中,点
,直线
,O'为等腰直角
关于直线l的“直角对称点”.
①当
时,写出点
的坐标______;
②连接
,求
长度的取值范围;
(2)⊙O的半径为8,点M是
上一点,以点M为直角顶点作等腰直角
,其中
,直线l与
、
分别交于E、F两点,同时
为等腰直角
关于直线l的“直角对称点”,连接
;当点M在
上运动时,直接写出
长度的最大值与最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f89deb952f57f4b3fa4887b098b7b91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/1/b2c8b870-e9fc-4241-a5b4-42d28e72ffbf.png?resizew=350)
(1)在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/967107eb8f4343c3e2b07896d9da656f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02ebce8b2a915356ed39f36c5bad2ebe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/866b81a8384cce4f24867baca2e6820c.png)
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5095a28bb1b91bf6bed9e2cfbd76bb18.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12fe32dfbd66709875c5b9f79c9496da.png)
②连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7fbae4adf65fd0d3e6159f6dde899.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0eb7fbae4adf65fd0d3e6159f6dde899.png)
(2)⊙O的半径为8,点M是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84a9dabb53dc826019fc8b6ae6d940c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e2dcd60283e6bf6fc034063ae076604.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/438f34bc8b04e8c494b91306ac6fe352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2dca049735b45fb9b2533c68605eddc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1da895d8bd043625a0839128252130d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84a9dabb53dc826019fc8b6ae6d940c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3319ff1fccda7edf5514db4bcbdb33c0.png)
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【推荐1】问题提出
(1)如图1,在
中,
于点
,
于点
,若
,
,求
的值;
问题探究
(2)如图2,在矩形
中,点
、
分别在边
、
上,连接
、
,且
.求证:
;
问题解决
(3)如图3,某地有一足够大的空地,现想在这片空地上修建一个平行四边形状的休闲区
,其中
,点
、
、
分别在边
、
、
上,管理部门欲从
到
、
到
分别修建小路,两条小路
、
交汇于点
,且满足
,
,为使美观现要沿平行四边形
的四条边修建绿化带(宽度忽略不计),求所修绿化带的长度(
的周长).
(1)如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e3f5dc11efe60b4fd9a13b1d6b83842.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fbee06d305abf6692125513dc3757f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f8eeeea1c9652cacce976f8129cf520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5ca1cff7e12ceae15df29736638545a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/8/93497f79-6a32-472f-a59a-2cae83c98cd0.png?resizew=743)
问题探究
(2)如图2,在矩形
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61fe81d0b136fc2acc97ab50ffbf6edf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c91cdc5ed01de035935279b4bdfea887.png)
问题解决
(3)如图3,某地有一足够大的空地,现想在这片空地上修建一个平行四边形状的休闲区
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fab0b6bae58dc0d07e87d7e1b3ea7423.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/907d5147cea4c9ce855074864fe54506.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e184f532aa6ff1209f9e9887513c2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1af33957f4ca4d51f3b7aeafc0bdbce6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
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【推荐2】如图一面墙上有一个矩形门ABCD现要打掉部分墙体将它改为一个圆弧形的门,在圆内接矩形ABCD中,AD
m,CD=1m.
(1)求此圆弧形门所在圆的半径是多少m?
(2)求要打掉墙体的面积是多少m2?(π≌3.1,
1.7,结果精确到1m2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ada26e9b7fe22be819d675f72e086ee.png)
(1)求此圆弧形门所在圆的半径是多少m?
(2)求要打掉墙体的面积是多少m2?(π≌3.1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/665c3b7da1695450500ee9711a24ebf7.png)
![](https://img.xkw.com/dksih/QBM/2021/5/2/2712445452484608/2714751934578688/STEM/4ab5107d-03c1-46d9-983f-bcf07ca2814b.png)
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【推荐1】在平面直角坐标系xOy中,给出如下定义:若点P在图形M上,点Q在图形N上,称线段PQ长度的最小值为图形M,N的密距,记为d(M,N),特别地,若图形M,N有公共点,规定d(M,N)=0.
![](https://img.xkw.com/dksih/QBM/2020/11/18/2595678965055488/2598609191264256/STEM/7d9e3993-93a9-4dc9-9f6c-6b22b32435cb.png?resizew=481)
(1)如图1,⊙O的半径为2,
①点A(0,1),B(4,3),则d(A,⊙O)= ,d(B,⊙O)=_______.
②已知直线l:y=
x+b与⊙O的密距d(l,⊙O)=
,求b的值.
(2)如图2,C为x轴正半轴上的一点,⊙C的半径为1,直线y=
x
与x轴交于点D,与y轴交于点E,其中∠ODE=30°,线段DE与⊙C的密距d(DE,⊙C)<
.请直接写出圆心C的横坐标m的取值范围.
![](https://img.xkw.com/dksih/QBM/2020/11/18/2595678965055488/2598609191264256/STEM/7d9e3993-93a9-4dc9-9f6c-6b22b32435cb.png?resizew=481)
(1)如图1,⊙O的半径为2,
①点A(0,1),B(4,3),则d(A,⊙O)= ,d(B,⊙O)=_______.
②已知直线l:y=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf7a8eccf88e7ec283f0389acc7c419e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3995e00757afc4db7fededf1b464574a.png)
(2)如图2,C为x轴正半轴上的一点,⊙C的半径为1,直线y=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08e91d2fa9519a5f48d488176700499.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32db98955bc6859d1895fa5e22bbd111.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/29092567e48972b3eae360e171d1ab10.png)
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【推荐2】已知二次函数图象的顶点在原点
,对称轴为
轴.一次函数
的图象与二次函数的图象交于
两点(
在
的左侧),且
点坐标为
.平行于
轴的直线
过
点.
(2)判断以线段AB为直径的圆与直线
的位置关系,并给出证明;
(3)把二次函数的图象向右平移2个单位,再向下平移t个单位(t>0),二次函数的图象与x轴交于M,N两点,一次函数图象交y轴于F点.当t为何值时,过F,M,N三点的圆的面积最小?最小面积是多少?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d053b14c8588eee2acbbe44fc37a6886.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6767830cc1811f0f4ea5a008fdc7e723.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fb398137779190b35492d9f06d5fbc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd107075b4df4c6e628808a3e51dfd99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a4a42e72592e26494de97efdf27323.png)
(2)判断以线段AB为直径的圆与直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(3)把二次函数的图象向右平移2个单位,再向下平移t个单位(t>0),二次函数的图象与x轴交于M,N两点,一次函数图象交y轴于F点.当t为何值时,过F,M,N三点的圆的面积最小?最小面积是多少?
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【推荐3】定义:在平面直角坐标系xOy中,点P为图形M上一点,点Q为图形N上一点.若存在OP=OQ,则称图形M与图形N关于原点O“平衡”.
(1)如图1,已知⊙A是以(1,0)为圆心,2为半径的圆,点C(﹣1,0),D(﹣2,1),E(3,2).
①在点C,D,E中,与⊙A关于原点O“平衡”的点是 ;
②点H为直线y=﹣x上一点,若点H与⊙A关于原点O“平衡”,求点H的横坐标的取值范围;
(2)如图2,已知图形G是以原点O为中心,边长为2的正方形.⊙K的圆心在x轴上,半径为2.若⊙K与图形G关于原点O“平衡”,请直接写出圆心K的横坐标的取值范围.
(1)如图1,已知⊙A是以(1,0)为圆心,2为半径的圆,点C(﹣1,0),D(﹣2,1),E(3,2).
①在点C,D,E中,与⊙A关于原点O“平衡”的点是 ;
②点H为直线y=﹣x上一点,若点H与⊙A关于原点O“平衡”,求点H的横坐标的取值范围;
(2)如图2,已知图形G是以原点O为中心,边长为2的正方形.⊙K的圆心在x轴上,半径为2.若⊙K与图形G关于原点O“平衡”,请直接写出圆心K的横坐标的取值范围.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/17/0a3dda4f-f971-4358-80d2-50e9611c6d45.png?resizew=446)
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