名校
1 . (1)问题发现
如图1,在Rt△ABC中,∠A=90°,
=1,点P是边BC上一动点(不与点B重合),∠PAD=90°,∠APD=∠B,连接 CD.
(1)①求
的值;②求∠ACD的度数.
(2)拓展探究
如图 2,在Rt△ABC中,∠A=90°,
=k.点P是边BC上一动点(不与点B重合),∠PAD=90°,∠APD=∠B,连接CD,请判断∠ACD与∠B 的数量关系以及PB与CD之间的数量关系,并说明理由.
(3)解决问题
如图 3,在△ABC中,∠B=45°,AB=4
,BC=12,P 是边BC上一动点(不与点B重合),∠PAD=∠BAC,∠APD=∠B,连接CD.若 PA=5,请直接写出CD的长.
如图1,在Rt△ABC中,∠A=90°,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea79586df2029edb34c7cb2f67dc3722.png)
(1)①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/897ded15cf9b0e0fb2400f17433be1e9.png)
(2)拓展探究
如图 2,在Rt△ABC中,∠A=90°,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea79586df2029edb34c7cb2f67dc3722.png)
(3)解决问题
如图 3,在△ABC中,∠B=45°,AB=4
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://img.xkw.com/dksih/QBM/2018/12/2/2088280646909952/2089582028005376/STEM/0ef3262715a1461d99b1f4f86ac9448a.png?resizew=376)
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2018-12-04更新
|
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9卷引用:陕西省西安市碑林区西安建筑科技大学附属中学2022-2023学年九年级上学期期中数学试题
陕西省西安市碑林区西安建筑科技大学附属中学2022-2023学年九年级上学期期中数学试题【校级联考】河南省南阳市新野县2018届九年级下学期第二次模拟考试数学试题河南省南阳市淅川县2019届九年级下学期第一次模拟测试数学试题河南省郑州市枫杨外国语2019-2020学年九年级上学期期中数学试题河南省郑州市高新区2019-2020学年九年级上学期期中数学试题(已下线)【万唯原创】相似三角形·基础必练 (一)(已下线)【万唯原创】河南省2020年中考数学-逆袭卷-特训22-23(已下线)【万唯原创】相似三角形·满分特训(二)广东省深圳市深圳实验学校光明部2022-2023学年九年级上学期10月月考数学试卷
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2 . 爱好思考的小茜在探究两条直线的位置关系查阅资料时,发现了“中垂三角形”,即两条中线互相垂直的三角形称为“中垂三角形”.如图(1)、图(2)、图(3)中,AM、BN是△ABC的中线,AM⊥BN于点P,像△ABC这样的三角形均为“中垂三角形”.设BC=a,AC=b,AB=c.
【特例探究】
(1)如图1,当tan∠PAB=1,c=4
时,a= ,b= ;
如图2,当∠PAB=30°,c=2时,a= ,b= ;
【归纳证明】
(2)请你观察(1)中的计算结果,猜想a2、b2、c2三者之间的关系,用等式表示出来,并利用图3证明你的结论.
【拓展证明】
(3)如图4,▱ABCD中,E、F分别是AD、BC的三等分点,且AD=3AE,BC=3BF,连接AF、BE、CE,且BE⊥CE于E,AF与BE相交点G,AD=3
,AB=3,求AF的长.
【特例探究】
(1)如图1,当tan∠PAB=1,c=4
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
如图2,当∠PAB=30°,c=2时,a= ,b= ;
【归纳证明】
(2)请你观察(1)中的计算结果,猜想a2、b2、c2三者之间的关系,用等式表示出来,并利用图3证明你的结论.
【拓展证明】
(3)如图4,▱ABCD中,E、F分别是AD、BC的三等分点,且AD=3AE,BC=3BF,连接AF、BE、CE,且BE⊥CE于E,AF与BE相交点G,AD=3
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://img.xkw.com/dksih/QBM/2017/8/17/1753978328113152/1753998513258496/STEM/974e7b203deb4b9ca63445a28dddf22c.png?resizew=478)
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2016-12-06更新
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8卷引用:2017年陕西省初中数学毕业学业模拟考试(一)
2017年陕西省初中数学毕业学业模拟考试(一)2017年陕西省中考数学模拟试卷(一)2016年初中毕业升学考试(湖北随州卷)数学(已下线)决胜2018中考压轴题全揭秘 专题16 四边形问题(已下线)决胜2018中考压轴题全揭秘 专题28 探究型问题(已下线)【万唯原创】2017年河南省中考数学-试题研究-第二部分解答题重难点突破题型62020年云大附中一二一校区九年级三模数学试题2022年广东省深圳市南山区九年级下学期第三次学情调研(三模)数学试题
3 . 【问题背景】
如图,
是边长为6的等边三角形,点
为
上一动点,连接
的垂直平分线分别交
于
,连接
.
【尝试证明】
(1)如图1,若点
是线段
上的动点.
求证:①
;
②
.
【拓展升华】
(2)如图2,若点
是线段
延长线上的点,当
时,求
与
的面积之比.
如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ff016af723be023e1525ce85d69fe94.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19409d6d120387b3bccd934c4911bb45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94165c5047e68e952df7a8dc6dc32fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/468848d6778ba56ef340bde4ff84834d.png)
【尝试证明】
(1)如图1,若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
求证:①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bb54d2b5fdd80b981bcc4e0797cc9f62.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7752dd084ab0a30d8febec991587ad9d.png)
【拓展升华】
(2)如图2,若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/714cc3707bba3bfdb56e251999be8592.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/276d0071f4fa02e931c31a1fc32d9618.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8289f505d2cfbec2a4264e14b18f946e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/21/c5eecd58-f90f-41be-8bf0-6d23aa44b9f9.png?resizew=262)
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4 . 【定义新知】
如图1,在线段
上有一点P,若
与
相似,则称点P为
与
的“似联点”.
【理解运用】
(1)如图2,在
的正方形网格中,四边形
的顶点均在格点上,连接
,在线段
上画出点P,连接
、
,使得点P为
与
的“似联点”;(只需画出一种情况)
(2)如图3,在
中,弦
与
相交于点P,连接
、
,试判断点P是否为
与
的“似联点”,并说明理由;
【拓展应用】
(3)如图4,现有一块四边形铁皮
,
,
,
,点E、F分别是
、
边上的定点,
,且
.工人师傅想在线段
上找出
与
的“似联点”P,并在点P处打孔,请你通过作图帮助工人师傅确定打孔的准确位置和数量(需说明理由),并求出孔(点P)与点E之间的距离.
如图1,在线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a855335176fc36a15017f50a8561348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96645a3530e72d5d733d2c72147d340.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a855335176fc36a15017f50a8561348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d96645a3530e72d5d733d2c72147d340.png)
【理解运用】
(1)如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13a96e9c2e2a15130d1d56a1d0e16b72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/520f8abda6a85e7ef6f281fc2df853fa.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/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a855335176fc36a15017f50a8561348.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03b465c5d8a5861b7b130e56dde8b1ae.png)
(2)如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3da7bcea5a45eeae211f5851f12a7517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/357265c532428e886a643e8e653eec9e.png)
【拓展应用】
(3)如图4,现有一块四边形铁皮
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47dfff00b8d24a52cbc4799d764c006.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c3614c5e351287527447998a15f749f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50ef47dd4b292de8efa931edf5580771.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/f96cd7f4747f277eb1f5dee6a644b1af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25eff69a4a0dc7a7ab183843303d333.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/028c33f1a5112a4e4cce0f0cb6a7cc3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c3a4638e2eb4e684b73416fb6a0f344.png)
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5 . 问题初探】:(1)如图①,在
中,点D、E分别在边
上,连接
.若
,则
的长为 ;
【问题深入】:(2)如图②,在扇形
中,点C是
上一动点,连接
求四边形
的面积的最大值;
【拓展应用】:(3)为进一步促进西安市文化和旅游高质量发展,推动全市文明旅游创建工作,结合2023年陕西省文明旅游示范单位申报工作,一并开展2023年西安市文明旅游示范单位评选工作.某地为参加评选积极改善环境,拟建一个四边形休闲广场
,其大致示意图如图③所示,其中
,
米.点E处设立一个自动售货机,点E是
的中点,连接
,
与
交于点M,连接
,沿
修建一条石子小路(宽度不计),将
和
进行绿化.根据设计要求,
.为倡导绿色新风尚,现要使绿化的面积尽可能的大,请问
和
的面积之和是否存在最大值?若存在,请求出
和
面积之和的最大值;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0d54431bbb28ebd98db5c1dc6083a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cce4398730ea72d5cd653ab1c5a4721.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a19338598965bb3856cdd0236bbf694.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
【问题深入】:(2)如图②,在扇形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4819c39c281427826e1b3f7a4c2b720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16d65cecaf8a3dc2953f4109c75a981e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0df0cdcfc8138d50c2e2a84774fcd92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ea16ceca816f7d3d50650af141baf42.png)
【拓展应用】:(3)为进一步促进西安市文化和旅游高质量发展,推动全市文明旅游创建工作,结合2023年陕西省文明旅游示范单位申报工作,一并开展2023年西安市文明旅游示范单位评选工作.某地为参加评选积极改善环境,拟建一个四边形休闲广场
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/902a133f004d0f09a04d05d44103c8b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87aab936ef1ddeaab717bc3d07a2c046.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/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc963e25d03b5c552bc063e44835364.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/376ca8f6e81ccd6846145ada511c4749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/765542d01be41196234ba5b3ec59a6c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc963e25d03b5c552bc063e44835364.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/376ca8f6e81ccd6846145ada511c4749.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc963e25d03b5c552bc063e44835364.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/376ca8f6e81ccd6846145ada511c4749.png)
您最近一年使用:0次
2024-04-24更新
|
107次组卷
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2卷引用:2024年陕西省宝鸡市陈仓区中考模拟数学试题
6 . 【基础巩固】
(1)如图1,在
中,点D、E分别在边
上,连接
,若
,求证:
;
【尝试应用】
(2)如图2,在
中,在边
上取一点E,以
为一边构造平行四边形
,使点D,F恰好落在边
上,连接
,若
,求
的长;
【拓展提升】
(3)如图3,在
中,在边
上取一点E,以
为一边构造平行四边形
,使点F恰好落在边
上,连接
,若
,求
的长.(提示:延长
交于点G)
(1)如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0d54431bbb28ebd98db5c1dc6083a75.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b8ee3c2174306999d94daf1fcea4a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1c5b29830e23ed86d888bb361c35147.png)
【尝试应用】
(2)如图2,在
![](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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08c14e87a2bcf7090eab2fea73667d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4515e1dff9a852b3294dc1d6488a5748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f425d058e864732a04caaeacf01917b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
【拓展提升】
(3)如图3,在
![](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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e08c14e87a2bcf7090eab2fea73667d2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f5c34f3a77590d035fe5c12c580cfd4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aa880102d854eaa020dafe59f101bc4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d8868d4d4ecb629a02439c890667777.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/22/1ef9e76f-70ee-468f-9c3b-2a1b33059737.png?resizew=377)
您最近一年使用:0次
名校
7 . 如图1,在
中,
,
,点
、
分别在边
、
上,
,连接
.将
绕点
顺时针方向旋转,记旋转角为
.
(1)[问题发现]
①当
时,
______;
②当
时,
的值是多少?请给出证明过程.
(2)[拓展研究]
试判断:当
时,
的大小有无变化?请仅就图2的情形给出证明;
(3)[问题解决]
在旋转过程中,
的最大值是多少?请直接写出答案.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/630d82ae0ed6deb825514e0bc92e74a0.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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62ceb3cc7e4c368c7b13a65d463f3c60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c24095e409b025db711f14be783a406c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/12/361973e9-54ef-4d87-982f-b05466adc897.png?resizew=421)
(1)[问题发现]
①当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/469451242301adbf795110cbb9154403.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc00342101809b43c1c3187dc0d06f9b.png)
②当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31cb3cc93e355ace2adee68be0a068f2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1672bf7cfa9a741651889f41855f6f3b.png)
(2)[拓展研究]
试判断:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8ef047d4e8392c80f84689e659a7a98e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1672bf7cfa9a741651889f41855f6f3b.png)
(3)[问题解决]
在旋转过程中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
您最近一年使用:0次
2023-12-09更新
|
97次组卷
|
6卷引用:陕西省西安市曲江第一中学2022-2023学年九年级上学期期中数学试题
8 . 【问题发现与证明】
如图①,正方形
中,
分别在边
、
上,且
,连接
,这种模型属于“半角模型”中的一类,在解决“半角模型”问题时,旋转是一种常用的分析思路.例如图中
与
可以看作绕点A旋转
的关系.这可以证明结论“
”,请补充辅助线的作法,并写出证明过程.
(1)延长
到点
,使
___________,连接
;
(2)求证:
.
【问题拓展与应用】
(3)某公园管理人员发现该公园有一块绿地,如图②所示,四边形
是平行四边形,已知
米,
米,
.为提升游客游览的体验感,准备修建三条赏花通道
、
、
,要求点
在
边上,点
为
边的中点,且
,现计划在
所在区域种植郁金香,种植郁金香的费用为每平方米12元,求该公园种植郁金香需要投入多少资金.
如图①,正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3cf9b288c48c73463a2f214f02b6952a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192155e6a3aade305b76b1eb7c75e30c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/441dec590b47adc3678a291a3ec89a4a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bc727c642cbc2181476b7dd8eca471e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c9af98a1a048b1e2f7c71e83c33498c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/7/2/3547703d-c03a-4b8f-98bf-3a72217e6a4d.png?resizew=405)
(1)延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/478d8cdad9506e889570ca6618a49615.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c9af98a1a048b1e2f7c71e83c33498c.png)
【问题拓展与应用】
(3)某公园管理人员发现该公园有一块绿地,如图②所示,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3dd4997da49b3dfe98144ad13aed0d5c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/667c50b897a3e209018f0ee09d37e73d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e075468e7fb0bf30229aec01a7205977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a9eb456c912e01458c19b06e83a5c9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
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9 . 问题初探:
如图1,已知
与
都是等腰三角形,顶角
,且
的中点均为
.请写出
与
间的数量关系,并证明;
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/11/649e84ed-e0d7-4701-b798-ddc2cc6b3f6f.png?resizew=499)
问题深入:
如图2,已知
与
都是等边三角形,
的中点均为
,请写出
与
间的数量关系,并证明;
拓展创新:
如图3,在
和
中,
,
,
,点
在
内部,直线
与
交于点
.直接写出线段
之间的数量关系________.
如图1,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed66b90ffe8c9a93aba917c61007e895.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c51424853857d7ca64d8b8dd74e972b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/11/649e84ed-e0d7-4701-b798-ddc2cc6b3f6f.png?resizew=499)
问题深入:
如图2,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c51424853857d7ca64d8b8dd74e972b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
拓展创新:
如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bdd83352f65446c38e118b0387fe7d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc75953bf5dcfa4af308c34bf9952d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7777a27259d26724229b604df42656c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/384cc52843b3e9b9c6089933b02aac1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.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/5240e1676825cb4573bd86891c623a60.png)
您最近一年使用:0次
2023-10-09更新
|
140次组卷
|
2卷引用:陕西省西安高新第一中学2023-2024学年九年级上学期第一次月考数学试题
22-23九年级上·浙江宁波·期末
10 . 【基础巩固】
(1)如图1,在四边形
中,对角线
平分
,
.求证:
;
【尝试应用】
(2)如图2,四边形
为平行四边形,
在
边上,
.点
在
延长线上,连结
,
,
,若
,
,
,求
的长;
【拓展提高】
(3)如图3,在
,
是
上一点,连结
,点
,
分别在
、
上,连结
,CE,
,若
,
,
,
,
,求
的值.
(1)如图1,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/645b06bbe036df96e4138c7732e45824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a64fb7d5e455a868ffa6ba46b6b19d1b.png)
【尝试应用】
(2)如图2,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c7e48723871f06a6aeae31a2a1ff79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41ff8c8d239cc12ddc3a899e9f054aa9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0215e13a9fb5574d5194aeb9507a98aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c4eccda4edcee29a5f15609e85106df.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8120f2eeb724c756b5f84a14c6df527.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f85e0f22b5bcb05d038d9bd172be097.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7cbfaec1d9dcaaf159b060163436113.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08e82a0052299f414cff1e7afcc5aa19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
【拓展提高】
(3)如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.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/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0215e13a9fb5574d5194aeb9507a98aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3620030e8808da46df97330103827913.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b2d0ac8b8bfe2f6f40c2cd4dacbe909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/662a9a633cdf3c5f19105025d69fc089.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e7c422a19b68b8b4cd484d22897e3a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/019080abcdd6bcf77ac8ecf57d77d84d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ca07b4b1d1d30ebf0a5a402ad8aeecf.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/22/120dbbc8-fc67-4425-8779-8e9f840f3e37.png?resizew=433)
您最近一年使用:0次
2023-08-13更新
|
179次组卷
|
6卷引用:陕西省榆林市定边县第七中学2023-2024学年九年级上学期月考数学试题