数学兴趣小组在活动时,老师提出了这样一个问题:
如图1,在
中,
,
,D是
的中点,求
边上的中线
的取值范围.
![](https://img.xkw.com/dksih/QBM/2023/11/8/3363539021922304/3383546090856448/STEM/c8b592bc01434cceb8db4a6e44cd4e13.png?resizew=531)
【阅读理解】小明在组内经过合作交流,得到了如下的解决方法:
(1)如图1,延长
,使
,连接
.根据__________可以判定
≌__________,得出
__________.
这样就能把线段
集中在
中.利用三角形三边的关系,即可得出中线
的取值范围是__________.
【方法感悟】当条件中出现“中点”,“中线”等条件时,可以考虑做“辅助线”——把中线延长一倍,构造全等三角形,把分散的已知条件和所求证的结论集中到同一个三角形中,这种作辅助线的方法称为“中线加倍”法.
【问题解决】(2)如图2,在
中,
,D是
边的中点,
,
交
于点E,
交
于点F,连接
,请判断
的数量关系,并说明理由.
【问题拓展】(3)如图3,
中,
,
,
是
的中线,
,
,且
,请直接写出
的长.
如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/addb8e20db1fbb40f17dea52f951b907.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://img.xkw.com/dksih/QBM/2023/11/8/3363539021922304/3383546090856448/STEM/c8b592bc01434cceb8db4a6e44cd4e13.png?resizew=531)
【阅读理解】小明在组内经过合作交流,得到了如下的解决方法:
(1)如图1,延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ad98ad714864041a632ca949308e417.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7ac5396c5ea442e0364b50c1db3d2da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae248960d8c1677cf948f8251275e863.png)
这样就能把线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/265264402dfa9999dafc0e6bfcfc94b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
【方法感悟】当条件中出现“中点”,“中线”等条件时,可以考虑做“辅助线”——把中线延长一倍,构造全等三角形,把分散的已知条件和所求证的结论集中到同一个三角形中,这种作辅助线的方法称为“中线加倍”法.
【问题解决】(2)如图2,在
![](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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b91f064bb73019053992fe781a53356a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b6677fb09f93db2caa6033861fcc1.png)
【问题拓展】(3)如图3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60d9142db4dd2ef151bf3d4a63afb61e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/531d6f90f144551a35d494b1fe7d2b89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8832c5450a61500ccbf73d95e16f449.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb64353c99068a7a1a8508a22f5b25b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
21-22八年级上·山东威海·期中 查看更多[17]
山东省威海市文登区文登第二中学2021-2022学年八年级上学期期中数学试题山东省威海市文登区2021-2022学年七年级上学期期中数学试题(已下线)专题12.34 作辅助线证明三角形全等-倍长中线(培优篇)(专项练习)-2022-2023学年八年级数学上册基础知识专项讲练(人教版)(已下线)第08讲 全等三角形的判定-ASA-【暑假自学课】2022年新八年级数学暑假精品课(人教版)山东省东营市河口区2021-2022学年七年级上学期期末数学试卷 (已下线)1.4 全等三角形的判定(ASA)(培优分阶练)-2022-2023学年八年级数学上册课后培优分级练(苏科版)(已下线)专题12.2.3 三角形全等的判定3(ASA)-【帮课堂】2022-2023学年八年级数学上册同步精品讲义(人教版)(已下线)重难点02 全等三角形(11种模型)-2022-2023学年八年级数学上学期考试满分全攻略(苏科版)吉林省长春市长春净月高新技术产业开发区2022-2023学年八年级上学期期末数学试题重庆市黔江区2022-2023学年八年级上学期期末考试数学试题(已下线)重难点02全等三角形中“倍长中线”模型-【暑假自学课】2023年新八年级数学暑假精品课(苏科版)(已下线)专题12.21 全等三角形几何模型(倍长中线)(分层练习)-2023-2024学年八年级数学上册基础知识专项突破讲与练(人教版)(已下线)专题1.21 全等三角形几何模型-倍长中线(分层练习)-2023-2024学年八年级数学上册基础知识专项突破讲与练(苏科版)(已下线)12.3(培优课)倍长中线(题型精讲精练)-【题型分类精粹】2023-2024学年八年级数学上学期期中期末复习讲练系列【考点闯关】(人教版)山东省淄博市张店区十一校联考2023-2024学年七年级上学期期中数学试题(已下线)专题04 构造全等三角形的四种方法-【好题汇编】备战2023-2024学年八年级数学上学期期末真题分类汇编(北京专用)2024年辽宁省大连市中考一模考前数学模拟预测题
更新时间:2023-12-06 20:25:03
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解答题-问答题
|
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(0.65)
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解题方法
【推荐1】(1)阅读理解:如图1,在△ABC中,若AB=10,BC=8.求AC边上的中线BD的取值范围,小聪同学是这样思考的:延长BD至E,使DE=BD,连接CE.利用全等将边AB转化到CE,在△BCE中利用三角形三边关系即可求出中线BD的取值范围,在这个过程中小聪同学证三角形全等用到的判定方法是 ;中线BD的取值范围是 .
(2)问题拓展:如图2,在△ABC中,点D是AC的中点,分别以AB,BC为直角边向△ABC外作等腰直角三角形ABM和等腰直角三角形BCN,其中∠ABM=∠NBC=90°,连接MN,探索BD与MN的关系,并说明理由.
(2)问题拓展:如图2,在△ABC中,点D是AC的中点,分别以AB,BC为直角边向△ABC外作等腰直角三角形ABM和等腰直角三角形BCN,其中∠ABM=∠NBC=90°,连接MN,探索BD与MN的关系,并说明理由.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/18/077d52eb-7eb5-4f3f-8347-5f85efd982eb.png?resizew=280)
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【推荐2】(1)如图①,在四边形
中,
,点
是
的中点,若
是
的平分线,试判断
,
,
之间的等量关系.
解决此问题可以用如下方法:延长
交
的延长线于点
,易证
得到
,从而把
,
,
转化在一个三角形中即可判断.
,
,
之间的等量关系________;
(2)问题探究:如图②,在四边形
中,
,
与
的延长线交于点
,点
是
的中点,若
是
的平分线,试探究
,
,
之间的等量关系,并证明你的结论.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b37591109b0a0ec5ffe2133f83310eca.png)
![](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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2947ca8e0cdbeb4aab80ce9e7b63ba98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
解决此问题可以用如下方法:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfebf927261bcc7963072daf3198cc40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2710d1ae999c3297ce2b89c93754a462.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
(2)问题探究:如图②,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b37591109b0a0ec5ffe2133f83310eca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac37366d2b54dc7d9a95ac6ddda5f3a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/28/a6c4766e-30b9-4b5e-89fe-7beddd6d3e41.png?resizew=245)
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【推荐1】如图,在梯形
中 ,
,
,
为
边的中点,
,
分别为
,
边上的点,若
,
,
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a34fdf9e6d2d87d01ad0bbb6a73ee05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78d14809daa5ee983530db8ea9105be2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f89deb952f57f4b3fa4887b098b7b91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb6cee83d36c4da913e0790e5070c46f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d64e0206b1814c35cc96bd2b6b12239a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5eba8bf0c4a8e49b3fac25832a0b0005.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0e57a13c665af88f326c9890072bf73.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/30/5fce5af0-7c4d-4f8a-86c1-d9be3a90d6c0.png?resizew=115)
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【推荐2】已知
ABC中∠BAC=130°,BC=18cm,AB、AC的垂直平分线分别交BC于E、F,与AB、AC分别交于点D、G.求:∠EAF的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce4cba95fc7d4853a243f8e3fb20ce70.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/19/493100b9-2cce-431f-b604-fd065a9fa6f5.png?resizew=180)
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【推荐1】已知
中,
,D是
边上一个动点,连接
,以
为直角边作等腰
,其中
.
(1)如图1,
①求证:
.
②线段
之间存在的数量关系为_________.
(2)如图2,若
,在动点D运动过程中,当
周长取得最小值时,求此时
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/149d217efaf5090696904003ec06393a.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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c2598f844644c4245eb226ecca57ee4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6eba7e2cb2e2390673b83bde332973d6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/10/37b943f2-f0ee-4831-ac38-25e4f17d5ac8.png?resizew=342)
(1)如图1,
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c503e29373e8d87134bdb46bd3912910.png)
②线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/800427bd0ab827a4b9891684b2aa705b.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a3c5d2cbe5cfa47fde68ff3b5b81469.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
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【推荐2】如图,在平面直角坐标系中,O为坐标原点,△ABO的边AB垂直于x轴,垂足为B,已知AB=BO=4.反比例函数
(k>0,x>0)的图象经过AO的中点C(2,2),交AB于点D.
(1)求反比例函数
的表达式;
(2)求经过C、D两点的直线所对应的函数表达式;
(3)设点E是x轴上的动点,请直接写出使△OCE为直角三角形的点E的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07854693dd2e33f66030d6106eb6e0ee.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/24/00d0a8d2-69f2-41fa-9a09-355116d8a87e.png?resizew=158)
(1)求反比例函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07854693dd2e33f66030d6106eb6e0ee.png)
(2)求经过C、D两点的直线所对应的函数表达式;
(3)设点E是x轴上的动点,请直接写出使△OCE为直角三角形的点E的坐标.
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