1 . 某数学兴趣小组在数学实践课上开展了“菱形折叠”研究活动.
第一步:每人制作边长都为7的菱形纸片若干个,四个顶点为A、B、C、D(为保持一致,活动中,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bca5f6366db8d915043332f818b65d7c.png)
小组内制作图形各点名称命名规则相同);
第二步∶在边
上分别取点M、N(不含端点),将四边形
沿
翻折,使线段
的对应线段
经过顶点 D(点A、B分别与点 E、F对应).
操作判断![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a96d1e67d5e74d6122c8dfa63f183.png)
则
______.
迁移探究
(2)缜密小组按上述步骤折叠后如图2所示,已知![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8861f0a5a206281022132864219f34d.png)
求
的长;
拓展延伸
(3)创新小组按上述步骤折叠后,要使
是以
为直角边的直角三角形,请你在图3 中帮他们画出满足条件的图形(草图即可),并求出对应的
的长.
第一步:每人制作边长都为7的菱形纸片若干个,四个顶点为A、B、C、D(为保持一致,活动中,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bca5f6366db8d915043332f818b65d7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c915b4ce31fabfd4703c547291ad9277.png)
第二步∶在边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ba48366317ebea1c9dd5e4e67e03092.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/baa76cfd79a2ff6b28f370e54f4eb51d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
操作判断
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a96d1e67d5e74d6122c8dfa63f183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c915b4ce31fabfd4703c547291ad9277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf86f352a109022b6a849690bd0def8b.png)
迁移探究
(2)缜密小组按上述步骤折叠后如图2所示,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8861f0a5a206281022132864219f34d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c915b4ce31fabfd4703c547291ad9277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
拓展延伸
(3)创新小组按上述步骤折叠后,要使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f44ca79d5e12bd02c393ba7f0b31c8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b46c607b3deac746c0ef3389ad8f65c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7785afeeaf274892253d04b4f693b367.png)
您最近一年使用:0次
2 . 直观感知:
(
)如图
,在四边形
中,
是等边三角形,
,
,将
绕点
顺时针旋转
得
,点
与点
重合,点
的对应点是点
.补全图形,并直接写出
的度数;
类比探究:
(
)如图
,在四边形
中,
,
,
,
,
,求
的长.
拓展运用:
(
)如图
,在四边形
中,
,
,
,
,
,在
的变化过程中时,求
的最大值.
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ec5a330d2c4a8bec71d1f9fb053aa04.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42fa2dd034894609a52abc94f499b3cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e3a9ec45721f7b4d1c99917ac0d970f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9524e3810e06dc781285f1289e75d653.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/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ea98cd15fa4c868eb83ca260c6d9ae6.png)
类比探究:
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a22b02f6c8636152aeb62d191251e931.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b10134e7a46e6f6f7cb9d5e2371727d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc11331a7b2d2619b40ee6d34c3bd620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b277f34aa35e65d6562e87050197ecda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
拓展运用:
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3202b1d9f838c32ab5765ce647d96b6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/846ee07de5ca5b23ae7d3fe240aca2c3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc11331a7b2d2619b40ee6d34c3bd620.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/430b77d8d2a99713b192dc729ddc2275.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
您最近一年使用:0次
3 . 【问题提出】
(1)如图①,在正方形
中,点E在
边上,连接
,
,垂足为点 G,交
于点 F.请判断
与
的数量关系,并说明理由.
【类比探究】
(2)如图②,在矩形
中,
,点E在
边上,连接
,
,垂足为点C,交
于点F.求
的值.
【拓展应用】
(3)如图③,在(2)的条件下,平移线段
,使它经过
的中点H,交
于点M,交
于点N,连接
,若
,则
的长为 .
(1)如图①,在正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/187ed13a7bd532bd39af5e5ad7493a2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
【类比探究】
(2)如图②,在矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/489d8955cfbbe8c0dc783d000c0beb88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/187ed13a7bd532bd39af5e5ad7493a2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a57799d26dbd66e5f21d327787a542e3.png)
【拓展应用】
(3)如图③,在(2)的条件下,平移线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.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/b5d8e33929752b1cb4dd36ee9b98b45d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcfc1e8e336020f326b7d3ee2ad7a306.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
您最近一年使用:0次
2024-04-22更新
|
200次组卷
|
6卷引用:2024 年青海省西宁市初中学考九年级调研测试(一模)数学试题
2024 年青海省西宁市初中学考九年级调研测试(一模)数学试题(已下线)重难点05 四边形压轴类型归纳(9大题型+满分技巧+限时分层检测)-2024年中考数学【热点·重点·难点】专练(广东专用)2024年四川省内江市威远县凤翔中学中考二模考试数学试题2024年山东省聊城市高唐县九年级下中考第二次模拟数学试题2024年山东省聊城市阳谷县九年级中考二模数学试题数学(黑龙江哈尔滨卷)-【试题猜想】2024年中考考前最后一卷
4 . A4纸是由国际标准化组织的
定义的,世界上多数国家所使用的纸张尺寸都是采用这一国际标准.这个标准最初是被魏玛共和国在1922年纳入
(编号是
),虽然其中一些格式法国在同一时期也自行研发出来,不过之后就被遗忘了.
定义了 A、B、C 三组纸张尺寸.
纸2次折叠,发现第1次的折痕与
纸较长的边重合,由此可求出
纸较长边与较短边的比为 .
(2)探究迁移;将一张
纸沿经过A、C两点的直线折叠,展开后得折痕
,再将其沿经过点B的直线折叠,使点A落在
上(O为两条折痕的交点),设第二条折痕与
交于点E.点E是否为
的中点?请说明理由.
(3)拓展应用;利用一张
纸经过裁剪获得一张边长为
的正方形纸片进行如下操作:对折正方形
得折痕
,连接
,将
折叠到
上,点B对应点H,得折痕
.试说明:G是
的黄金分割点.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51f8c9878aa38e66292c08b2c2cd4f68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce410af12783743652e99efa82299f16.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4753f86e979155e02940fbdf82f1e028.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51f8c9878aa38e66292c08b2c2cd4f68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b553e2a281714677e29a5e97dd7d5a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b553e2a281714677e29a5e97dd7d5a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b553e2a281714677e29a5e97dd7d5a3.png)
(2)探究迁移;将一张
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b553e2a281714677e29a5e97dd7d5a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4086335b5413c188dbcbb45484a1f07e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
(3)拓展应用;利用一张
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b553e2a281714677e29a5e97dd7d5a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a835d77c364303ea59d96e6ce42a58.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
您最近一年使用:0次
2024-04-19更新
|
208次组卷
|
5卷引用:2024年江苏省徐州市睢宁县第二中学中考模拟数学模拟预测题(4月)
2024年江苏省徐州市睢宁县第二中学中考模拟数学模拟预测题(4月)2024年贵州省黔东南州九年级数学中考模拟测试卷(一)(已下线)重难点05 几何压轴综合(8大题型+满分技巧+限时分层检测)-2024年中考数学【热点·重点·难点】专练(上海专用)数学(长春卷)-【试题猜想】2024年中考考前最后一卷2024年江苏省连云港市海州区连云港外国语学校中考二模数学试题
5 . 【背景阅读】我国古代著名数学著作《周髀算经》记载了“勾三、股四、弦五”,直观地证明了勾股定理,我们把三边的比为
的三角形称为
型三角形,例如:三边长分别为9,12,15的三角形就是
型三角形.
【实践操作】如图1,在正方形纸片
中,
,点E为边
上的中点,将
沿
折叠得
,延长
交
于点G,交
的延长线于点H.
【问题解决】(1)证明
是
型三角形;
(2)在不添加字母的情况下,直接写出图1中还有哪些三角形是
型三角形;
【拓展探究】(3)如图2,在矩形纸片
中,
,
,E是
上的一点,将
沿
折叠得到
,延长
交
于点G.其中
是
型三角形,请求出
的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4d44d5798d856568f7f6e5e91269ad5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6658c123fcd1668533d39bdea1787fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6658c123fcd1668533d39bdea1787fa.png)
【实践操作】如图1,在正方形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca2698f08d94afa1cd5b81444791f4c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
【问题解决】(1)证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47ffab24578093dee05cc834c02a8023.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6658c123fcd1668533d39bdea1787fa.png)
(2)在不添加字母的情况下,直接写出图1中还有哪些三角形是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6658c123fcd1668533d39bdea1787fa.png)
【拓展探究】(3)如图2,在矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca2698f08d94afa1cd5b81444791f4c7.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/36e5e61804ce550636a0354e0a78a22d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6658c123fcd1668533d39bdea1787fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e5e61804ce550636a0354e0a78a22d.png)
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6 . 某数学学习小组在学习旋转相关知识后,对特殊的四边形进行探究,有如下深究过程.
【问题解决】
(1)如图①,在矩形
中,点
为
边上一点,将
绕点
顺时针笑转90°后得
.若点
恰好落在
边上,求证:
;
【问题探究】
(2)如图②,在正方形
中,点
为
的中点,将
绕点
原时针旋转90°后得
.连接
,
.若
,求点
到
的距离;
【拓展延伸】
(3)如图③,在菱形
中,点
为
边上任意一点,点
在
上,
.
,
交于点
.若
,
,当
为等腰三角形时,直接写出
的长.
【问题解决】
(1)如图①,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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.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/32aa96520f55c12f723ef9b9beefb8c2.png)
【问题探究】
(2)如图②,在正方形
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
【拓展延伸】
(3)如图③,在菱形
![](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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d1b54e7477a5960285b14fb753caf87.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/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4494a85de0be0b97a69348115aef8513.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/454328a8e75953fdb0835ce80d9566e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
您最近一年使用:0次
7 . 【定义呈现】有两个内角分别是它们对角的两倍的四边形叫做倍对角四边形,其中,这两个内角称为倍角.例如:如图1,在四边形
中,
,
,那么我们就叫这个四边形是倍对角四边形,其中
,
称为倍角.
【定义理解】如图1,四边形
是倍对角四边形,且
,
是倍角.求
的度数;
【拓展提升】如图2,四边形
是倍对角四边形,且
,
是倍角,延长
、
交于点A.在
下方作等边三角形
,延长
、
交于点G.若
,
,
,四边形
的周长记为
.
的代数式表示
;
(2)如图3,把题中的“
”条件舍去,其它条件不变.
①求证:
;
②探究
是否为定值.如果是定值,求这个定值,如果不是,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27bb421b8fb47716b09ea7367a8a6a42.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/525b212fc8e4a6e4e3238ef3037831e8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c543c3ddc3723fde6bbfca3ea3b921b.png)
【定义理解】如图1,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c543c3ddc3723fde6bbfca3ea3b921b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4453fb13033ef0e1547a41ed973940c7.png)
【拓展提升】如图2,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1333acc72211e3ddb9a0f8c726ce8a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c5d3f1c7f7ffdf572978468d5a3adde.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d3e081637bcea5368cc72370bae283b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6830ebecddbd9759be626289c408e4f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/735056c174e8dd7906257a2a50a962a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef0402dd5ae3db10281f9f1e11738bcb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15d274d36400ca59460848b59307979c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1333acc72211e3ddb9a0f8c726ce8a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0a532e15e232cb4b99a8d4d07c89575.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
(2)如图3,把题中的“
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
①求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d707c196e6cab04a80151d542de9ee46.png)
②探究
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41bd21c7396562410aac59dfe6f01321.png)
您最近一年使用:0次
8 . 某数学兴趣小组在探究“手拉手”模型时,等边三角形
和
按如图1摆放,连接
延长
交
于点F,连接
,保持
不动,将
绕点A旋转.
,
重合时,请直接写出
之间的数量关系:______;
【深入探究】(2)如图1,当点E,F不重合时,(1)中的结论是否仍然成立?若成立,请给出推理过程;若不成立,请说明理由.
【拓展延伸】(3)如图3,当
和
都是等腰直角三角形,
.连接
延长
交
于点F,连接
,试探究
之间的数量关系,并说明理由.
【推广应用】(4)如图4,在
中,若
.连接
延长
交
于点F,连接
,请直接写出
之间的数量关系:______;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d24d4d6370103ee189a09a1e21597eb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b41d4070854edfaa24071137b314cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08a50b9dbe5e8c960fb848f425006984.png)
【深入探究】(2)如图1,当点E,F不重合时,(1)中的结论是否仍然成立?若成立,请给出推理过程;若不成立,请说明理由.
【拓展延伸】(3)如图3,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21c2425db34d83dbe44f3ee3d2c5439b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40b72e2082bb7bada0b7fc433723b78a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b41d4070854edfaa24071137b314cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08a50b9dbe5e8c960fb848f425006984.png)
【推广应用】(4)如图4,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed8403902de9e7e6e428e25b8a24126e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6fad52bca7ff0186960747ac45e7a21c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40b72e2082bb7bada0b7fc433723b78a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b41d4070854edfaa24071137b314cb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08a50b9dbe5e8c960fb848f425006984.png)
您最近一年使用:0次
2023-11-20更新
|
134次组卷
|
2卷引用:湖北省省直辖县级行政单位潜江市初中12校联考2023-2024学年九年级上学期月考数学试题
9 . 问题背景:如图1,在四边形
中,
,将
沿
翻折,点
的对应点
恰好落在
边上.
连接
,判断
的形状,说明理由;
(2)探究迁移
将
沿射线
平移得到
(点
的对应点分别为
),当点
的对应点
与点
重合时,求四边形
的周长;
(3)拓展创新
将
继续沿射线
平移得到
(点
的对应点分别为
),
与
交于点
,且
,将
绕点
在平面内自由旋转,当
时,直接写出
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1db7d85a24547dde9e8ec84928eb09e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
(2)探究迁移
将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ec75f5ccd5756d5b787c061c831dffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cb452339f63edb5369b899addce5b68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de4aae7fbed0dc38b911aaab38a9534.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e33d53d6c60a069de0c42ebcf5cb87ee.png)
(3)拓展创新
将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ec75f5ccd5756d5b787c061c831dffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cb452339f63edb5369b899addce5b68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de4aae7fbed0dc38b911aaab38a9534.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24cfad1ba71a78d8f415335cde2f8c52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0435f724a67b42436680115721cc3f1d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ec75f5ccd5756d5b787c061c831dffd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1d0a81a42c37ef1128ccf1ea646bb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6024fd4532f5f981deac4582c799a6ef.png)
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10 . 在数学活动课中,老师组织学生开展“如何通过折纸的方法,确定矩形纸片长边上的一个三等分点”的探究活动.
【操作探究】
“求知”小组经过一番思考和讨论交流后,进行了如下操作,如图1.
第1步:先将矩形纸片
沿对角线
对折,展开铺平,折痕为
;
第2步:将边
以某一合适长度向右翻折3次,折痕
与
交于点K;
第3步:过点K折叠矩形纸片,使折痕
,
交
于点N;
第4步:延长
交边
于点P,则点P为边
的三等分点.
证明过程如下:
由题意,得
.
∵
,∴
.
∴① .
∴
.同理,得
.
∴② .
∴
.则点P为边
的三等分点.
“励志”小组的操作如下,如图2.
第1步:先将矩形纸片
沿对角线
对折,展开铺平,折痕为
;
第2步:再将矩形纸片对折,使点A和点B重合,展开铺平,折痕为
;
第3步:沿
折叠矩形纸片,折痕
交
于点G;
第4步:过点G折叠矩形纸片,使折痕
.
【过程思考】
(1)补全“求知”小组证明过程中①②所缺的内容;
(2)“励志”小组经过上述操作,认为点M为边
的三等分点.请你判断“励志”小组的结论是否正确,并说明理由.
【拓展应用】
(3)如图3,将矩形纸片
对折,使点A和点B重合,展开铺平,折痕为
,将边
沿
翻折到
的位置,过点G折叠矩形纸片,使折痕
,若点M为边
的三等分点,求
的值.
【操作探究】
“求知”小组经过一番思考和讨论交流后,进行了如下操作,如图1.
第1步:先将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
第2步:将边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71432131d194730a67f3192d1f8679b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
第3步:过点K折叠矩形纸片,使折痕
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4bd9ab622a02ba751f565f749576bf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bc0f6aa0b8ec19dde4db21b3d214d0e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第4步:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
证明过程如下:
由题意,得
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/950476c26e07a8c863453d1d9b79c8c3.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4bd9ab622a02ba751f565f749576bf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eca58c00cfbf0d39b94e0f9858bb9619.png)
∴① .
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fb8b8bb5991ad048f810faff978b6b0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc43db651de835c0b32a0161d24ee530.png)
∴② .
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/200a0f17361099266c313af5e2362258.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
“励志”小组的操作如下,如图2.
第1步:先将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
第2步:再将矩形纸片对折,使点A和点B重合,展开铺平,折痕为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第3步:沿
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
第4步:过点G折叠矩形纸片,使折痕
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/198beeaef815604729e37c09b24da353.png)
【过程思考】
(1)补全“求知”小组证明过程中①②所缺的内容;
(2)“励志”小组经过上述操作,认为点M为边
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
【拓展应用】
(3)如图3,将矩形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](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)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/198beeaef815604729e37c09b24da353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2465ee1fec12351943f15055dfd6ec73.png)
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