1 . 课本再现
如图1,四边形
是菱形,
,
.
(1)求
的长.
应用拓展
(2)如图2,
为
上一动点,连接
,将
绕点
逆时针旋转
,得到
,连接
.
①直接写出点
到
距离的最小值;
②如图3,连接
,若
的面积为6,求
的长.
如图1,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ab5c45a72849d2cae1d65b282b5bd19.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/454328a8e75953fdb0835ce80d9566e3.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dec2ca6438c82b43f746057d8129885.png)
应用拓展
(2)如图2,
![](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/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6c0927afc571a7c966c98192040979e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
①直接写出点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
②如图3,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9e1ff2308235f7109bf67a59113828b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20bb77c849e0ce2a7d06ff0ca900fb28.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
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2 . 如图1,在正方形
中,
,E是对角线
上一动点(不与点A、C重合),连接
,作
交边
或边
的延长线于点F,以
和
为邻边构造矩形
,连接
.
的数量关系是______;位置关系是______.
(2)如图2,当
时,求
的长.
(3)设
,求y与x之间的函数解析式.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6c6e7c025362c46a64a8956761f08e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f0805cc7a96c866df10b667ac4851e.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/21037e170bdbb322558e79c40c00b454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24f515246d9ee093bfbf9983fd47357d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/508dba752f8b01895a4d1788e8f67fb0.png)
(2)如图2,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047eba5bba202f75349f94ad9feb4962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f1f05d924a2993fa31325a9447f30e7.png)
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3卷引用:江西省南昌市南昌县2023-2024学年八年级下学期第二次月考数学试题
江西省南昌市南昌县2023-2024学年八年级下学期第二次月考数学试题安徽省芜湖市无为市部分学校2023-2024学年八年级下学期月考数学试题(已下线)暑假作业07 函数的基础概念(4大题型巩固提升练+拓展能力练+仿真考场练)-【暑假分层作业】2024年八年级数学暑假培优练(人教版)
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3 . 如图,直线
上有两点
、
,分别引两条射线
、
,
,
,射线
、
分别绕
点,
点以
度/秒和
度/秒的速度同时顺时针转动,在射线
转动一周的时间内,使得
与
平行所有满足条件的时间![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
___________ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ceb8ac3f3b2598a84e0bf85b713ecdb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/348e40cf88acadb42e0d696fdc30dd2d.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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
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4卷引用:江西省赣州市瑞金市2023-2024学年七年级下学期期中数学试题
江西省赣州市瑞金市2023-2024学年七年级下学期期中数学试题山东省济南市历下区实验初级中学2023-2024学年七年级下学期期中数学试题(已下线)期中各名校真题-压轴必刷题-2023-2024学年七年级数学下学期期中期末考点归纳满分攻略讲练(人教版)(已下线)暑假作业02 平行线的判定与性质(知识梳理+三大题型专练+能力拓展练)-【暑假分层作业】2024年七年级数学暑假培优练(人教版)
4 . 如图,
是
的直径,
,点C在线段
上运动,过点C的弦
,将
沿
翻折交直线
于点F,当
的长为正整数时,线段
的长为______ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c38dfd14dde969702dff97ef2270f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ff3d1126edfb95418994f607db08ff0.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/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2319a01218514917e446dfc807a625ff.png)
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5 . (1)某学习小组在探究三角形全等时,发现了下面这种典型的基本图形.如图1,已知:在
中,
,
,直线l经过点A,
直线l,
直线l,垂足分别为点D、E.证明:
.
(2)组员小刘想,如果三个角不是直角,那结论是否会成立呢?如图2,将(1)中的条件改为:在
中,
,D、A、E三点都在直线l上,并且有
,其中α为任意锐角或钝角.请问结论
是否成立?如成立,请你给出证明;若不成立,请说明理由.
(3)数学老师赞赏了他们的探索精神,并鼓励他们运用这个知识来解决问题:如图3,过
的边
、
向外作正方形
和正方形
,
是
边上的高,延长
交
于点I,求证:I是
的中点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c06154cae3bf7a8ce5a1e97a7380875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a5928c98b341b16d4b5a5b931d2929d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44b190c8d3d7d7d0e6e959e8a52eae90.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb773a664fa72fc5d8fe377e9f891901.png)
(2)组员小刘想,如果三个角不是直角,那结论是否会成立呢?如图2,将(1)中的条件改为:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cf8edbd31ff7e27c784c8688b537a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb773a664fa72fc5d8fe377e9f891901.png)
(3)数学老师赞赏了他们的探索精神,并鼓励他们运用这个知识来解决问题:如图3,过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.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/ad3a079cfdcca9acdacecbf08f9f78cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff7a0bfc593a8a33b6cade6ba213904c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/184921a45f87423e92cf1e627ad55143.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
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6 . 已知在
中,
,点D为
边上一动点,以
为边,在
的右侧作等边三角形
.
平分
时,四边形
是______形.
(2)如图(2),过点E作
于点F,
与
具有怎样的关系?F为
的中点吗?说明理由.可得出结论,无论运动到何处,点E在
的何处?
(3)如图(3),若
,利用(2)中结论.
①当D为
的中点时,过点E作
于点G,求
的长;
②点D从点B运动到点C,则点E所经过的路径长为多少?
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/052b54fe911c39672315682c7a7e6b56.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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ea91137f4427a67f3a8783d974ef7e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bc1cbcd6d00f0c36bad8254297d9f33.png)
(2)如图(2),过点E作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8de4a54cc7818be87a239f6de5f5d05b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
(3)如图(3),若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
①当D为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8c13ab7f17263c4e6163d960f1c5060.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
②点D从点B运动到点C,则点E所经过的路径长为多少?
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7 . 综合与实践
在一次综合实践活动课上,王老师给每位同学各发了一张正方形纸片,请同学们思考如何仅通过折纸的方法来确定正方形一边上的一个三等分点.
【操作探究】
“乘风”小组的同学经过一番思考和讨论交流后,进行了如下操作:
第1步:如图1所示,先将正方形纸片
对折,使点A与点B重合,然后展开铺平,折痕为
;
第2步:将
边沿
解析到
的位置;
第3步:延长
交
于点
,则点
为
边的三等分点.
“破浪”小组是这样操作的:
第1步:如图2所示,先将正方形纸片对折,使点A与点
重合,然后展开铺平,折痕为EF;
第2步:再将正方形纸片对折,使点
与点
重合,再展开铺平,折痕为
,沿
翻折得折痕
交
于点
;
第3步:过点
折叠正方形纸片
,使折痕
.
(1)“乘风”小组的证明过程中,①处的推理依据是_______;②处的方程是_______;
(2)结合“破浪”小组操作过程,判断点
是否为
边的三等分点,并证明你的结论;
【拓展提升】
(3)①如图3,将矩形纸片
对折,使点A和点
重合,展开铺平,折痕为
,将
沿
翻折得到
,过点
折叠矩形纸片,使折痕
,若点
为边
的三等分点,请求出
的值.
②在边长为3的正方形
中,点
是射线
上一动点,连接
,将
沿
翻折得到
,直线
与直线
交于点
.若
,请直接写出
的长.
在一次综合实践活动课上,王老师给每位同学各发了一张正方形纸片,请同学们思考如何仅通过折纸的方法来确定正方形一边上的一个三等分点.
【操作探究】
“乘风”小组的同学经过一番思考和讨论交流后,进行了如下操作:
第1步:如图1所示,先将正方形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第2步:将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e5f736b1195fef1d2d300168a795f1.png)
第3步:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
证明过程如下:连接![]() ![]() ![]() ![]() ![]() 又 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() 解得: ![]() ![]() ![]() |
第1步:如图2所示,先将正方形纸片对折,使点A与点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
第2步:再将正方形纸片对折,使点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
第3步:过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff3c1ea812dbddf9c5752e45e3713a7.png)
(1)“乘风”小组的证明过程中,①处的推理依据是_______;②处的方程是_______;
(2)结合“破浪”小组操作过程,判断点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
【拓展提升】
(3)①如图3,将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6818a98204f62c1b16699d26ca0c3f62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b31d7cff6b825f7abd8765e4cb289e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff3c1ea812dbddf9c5752e45e3713a7.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/2465ee1fec12351943f15055dfd6ec73.png)
②在边长为3的正方形
![](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/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6818a98204f62c1b16699d26ca0c3f62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b31d7cff6b825f7abd8765e4cb289e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9137125c2c2f7f262f7c8a9216451458.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
您最近一年使用:0次
名校
8 . 如图,已知抛物线
:
与y轴相交于点C,顶点为D.
(2)点P为直线CD左上方抛物线上的一动点,过点P作y轴的平行线交直线CD于点Q,当线段PQ取得最大值时,在抛物线的对称轴上找一点G,使
的周长最小,求点G的坐标;
(3)将抛物线
向左平移2个单位长度得到抛物线
,
与
相交于点E,点F为抛物线
对称轴上的一点,在平面直角坐标系中是否存在点H,使以点C,E,F,H为顶点的四边形为菱形,若存在,请直接写出点H的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c34081d1ee5e18cb9c3f6b324553349f.png)
(2)点P为直线CD左上方抛物线上的一动点,过点P作y轴的平行线交直线CD于点Q,当线段PQ取得最大值时,在抛物线的对称轴上找一点G,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f12a227f113313bdd42728e666e4614.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/23f3ffe7abc59e2f65d827c8eab8d36a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
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9 . 【问题背景】如图,正方形
的对角线相交于点O,点O又是正方形
的一个顶点,而且这两个正方形的边长相等,无论正方形
绕点O怎样转动,两个正方形重叠部分的面积,总等于一个正方形面积的
,九年级数学兴趣小组对上面的问题又进行了拓展探究、内容如下:正方形
的对角线相交于点O,点P落在线段
上,
(k为常数).
(1)如图1,将
的直角顶点P与点O重合,两直角边分别与边
,
相交于点M,N.
①填空:
______;
②求证:
.
【类比探究】
(2)如图2,将图1中的
沿
方向平移,判断
与
的数量关系(用含k的式子表示),并说明理由.
【拓展运用】
(3)如图3,点N在边
上,
,延长
交边
于点E,若
,求k的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4635a4e561d862638f6818e85e74bfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4635a4e561d862638f6818e85e74bfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d266a04f3dc7483eddbc26c5e487db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d52f125abf9625888d5feb7d6d2b5fd7.png)
(1)如图1,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d244880e9a28a02df25b4e864b28975e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
①填空:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd707b69a11f8de5566f23c1a2a9ff5a.png)
②求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4acd79bb9fb06f7c806eb6e17e4b613.png)
【类比探究】
(2)如图2,将图1中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91b51d3992644d37dc71c9b5a97d515c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6c8ffe24cf9f327aeb241225ab15ab1a.png)
【拓展运用】
(3)如图3,点N在边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f3f27564bb46555b27e130fe9aad7615.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ffc7d1af9053b027cf9e726f5367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f3d9105c0a55832a1ba674ec12364b1.png)
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10 . 【课本再现】“直角三角形斜边上的中线等于斜边的一半”是直角三角形的一条重要性质定理.如图1,在
中,
,点D是
的中点.求证:
.
下面是两位同学两种添加辅助线的方法:
小明:如图2,延长
至点E,使
,连接
;
小华:如图3,取
的中点E,连接
;
(1)请你选择其中一位同学的方法完成证明,聪明的你也可以利用图1用其他方法完成证明.
中,
是高,求证:B,C,D,E四点共圆.
【拓展提升】(3)如图5,在五边形
中,
,
,F为
的中点,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c71622531dfa894f21b2da123d020d24.png)
下面是两位同学两种添加辅助线的方法:
小明:如图2,延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0fa1aa5a7a5bb172ed4603f17c8b2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bba87bb6ab3a88f6d9529e01ce585a5d.png)
小华:如图3,取
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
(1)请你选择其中一位同学的方法完成证明,聪明的你也可以利用图1用其他方法完成证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/692c58a2af64d705cd4a988ed2bfbc3d.png)
【拓展提升】(3)如图5,在五边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9142a8490de14a87eda628ffa7e28982.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/766ae8acaddb28f8a5a55eff086fd976.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd34a9d150aff3aa789230d7772384a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/193ea44749f1c64c8723e84a57d15cb9.png)
您最近一年使用:0次