1 . 如图1,点
是正方形
两对角线的交点,分别延长
到点
,
到点
,使
,
,然后以
、
为邻边作正方形
,连接
,
.
;
(2)如图2,正方形
固定,将正方形
绕点
逆时针旋转
角(
),得到正方形
;
①在旋转过程中,当
是直角时,求
的度数;
②若正方形
的边长为2,在旋转过程中,
长的最大值为______.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4e16ead7fdee4751e871187dd85843e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bab4646c1695b78fc09d8287d3453c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/307bd991211ec79b47a4be52933bb8e7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b670c6d27ce68f17ac12c4efa1933db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbd99874843902d155f3f9dc26b924f1.png)
(2)如图2,正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b670c6d27ce68f17ac12c4efa1933db1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/783d872617eed2bbd474035156b6caaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e837d949ac97755fb0d52b1837120f1.png)
①在旋转过程中,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d3010872e39c34cb5fe799a44eb850b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
②若正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cad0a19415e796564f30906f2e7dbf76.png)
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2 . 综合与实践:
数学模型可以用来解决一类问题,是数学应用的基本途径.通过探究图形的变化规律,再结合其他数学知识的内在联系,最终可以获得宝贵的数学经验,并将其运用到更广阔的数学天地.
如图1,在
和
中,
,
,
,连接
,
,延长BE交
于点D.则
与
的数量关系: ,
.
(2)类比探究:
如图2,
和
均为等腰直角三角形,
,连接
,
,且点B,E,F在一条直线上,过点A作
,垂足为点M.请猜想
,
,
之间的数量关系,并说明理由;
(3)实践应用:
如图3,正方形
中,
,M点为线段
中点.将正方形
绕点A顺时针旋转,形成正方形
.连接
、
,直线
交直线
于点P,则线段
最大值为 .
数学模型可以用来解决一类问题,是数学应用的基本途径.通过探究图形的变化规律,再结合其他数学知识的内在联系,最终可以获得宝贵的数学经验,并将其运用到更广阔的数学天地.
如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d38d97f03faed3152db2fd3bd1919944.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1d0dd547921b8a35acf7b407fac21b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9b5c0e0724cce5c25a88dca5e9220c2.png)
(2)类比探究:
如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a93128719573a60a1bdcf0ede1116b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/396632dba26d326a6b69f0bb7016bd14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
(3)实践应用:
如图3,正方形
![](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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdd511e3eb724a9add64021b9b6f8c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b25f3ea33cc08b1e2a0d9c3a9dccaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6655cc150ddc9deba2254780984d0024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b25f3ea33cc08b1e2a0d9c3a9dccaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6655cc150ddc9deba2254780984d0024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
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名校
3 . 已知正方形
,
是
边上一点,过点
作
交
的延长线于点
,交
的延长线于点
.
,则线段
之间有怎样的数量关系: (直接写出结论);
(2)如图2,如果过点
作
交
的延长线于点
,那么
,请说明理由;
(3)如图3,在(2)的条件下,过点
作
于点
,连接
,若
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9bd33dcb66a0a342ec072456dac06659.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.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/b5c2293f93791a597bf0162411f3395f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1a6e42aa77ba929cb1384211f6e564.png)
(2)如图2,如果过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3f6141b4d267187f855681592b16a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d97dc3b752832906de41447bb58a341.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/875e60688276eba6b0b9235b43257bbe.png)
(3)如图3,在(2)的条件下,过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2f1054e3a5c0fc7dca3d285108112c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7bb1b34ba0a8c480f55dad078456663.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
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4 . 如图,
是正方形
外一点,连接
交
于点
.连接
,
,且
.
与
之间的数量关系,并说明理由;
(2)求证:
平分
;
(3)如备用图,过点
作
于点
,分别交
,
于点
,
,连接
,若
.求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a241cddccc81682fbdb0cee24687ed0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d29e7c4d6216690152430b09f9cf0c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/23dc2d2dd56fcc67698c45a6e0e48f80.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc56c77464a17a1e97b568762a3e2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb3de20ba8bda63c650ef571be155702.png)
(3)如备用图,过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55e45b9efd79e3e9e9101cf0eb98c824.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8861f0a5a206281022132864219f34d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cbf607284227006fc3b68074ffe31a2.png)
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名校
5 . 【教材回顾】
(1)苏科版教材八下第九章《中心对称图形—平行四边形》习题中有这样的问题:如图1,
的顶点 O在正方形
两条对角线的交点处,
, 将
绕点O旋转,
的两边分别与正方形
的边边
和
交于点
和点
(点
与点
,
不重合),问:在
旋转过程中,
与
具有怎样的数量关系?
爱思考的小歆和小涵同学分别探究出了如下两种解题思路:
小歆:考虑到正方形对角线的特征,过点O分别作
于点G,
于点H,即可通过证明三角形全等得到
与
的数量关系.
小涵:利用正方形对角线垂直、相等且互相平分等性质证明了三角形全等,可以得到
与
的数量关系.
通过他们的思路点拨,你认为
与
的数量关系为 ,并请选择一种思路去证明;
【类比探究】
(2) 如图2, 若将(1) 中的“正方形
”改为“
的菱形
”, 其他条件不变,当
时,判断以下结论正确的有 (填写所有正确的结论序号),并选择一个正确的结论去证明.
①
; ②
;
③四边形
的周长为定值; ④四边形
的面积为定值.
【拓展应用】
(3) 如图3, 学校内有一块四边形的花圃
, 满足
,
,
, 花圃内铺设了一条小路
,
平分
, 为方便学生赏花, 现计划修建一条径直的通道
与小路
相连,且
,入口点E恰好在
的延长线上.直接写出入口到点 A 的距离
的长 .
(1)苏科版教材八下第九章《中心对称图形—平行四边形》习题中有这样的问题:如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b2557547eecdc826b8282058802d217.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/319ff9f648fedec284a11a9761cb6183.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b2557547eecdc826b8282058802d217.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b2557547eecdc826b8282058802d217.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/2b2557547eecdc826b8282058802d217.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
爱思考的小歆和小涵同学分别探究出了如下两种解题思路:
小歆:考虑到正方形对角线的特征,过点O分别作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbdaf6769ca790186718cc60c7b89ba6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27acaeba896dd1a4a221151ffa464848.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
小涵:利用正方形对角线垂直、相等且互相平分等性质证明了三角形全等,可以得到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
通过他们的思路点拨,你认为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
【类比探究】
(2) 如图2, 若将(1) 中的“正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5b0382c28547d3834ca71f3f0677695.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd9b4bb7c5f4a856c3e0ca747fd64fb9.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcf2e3236ea30ee2c37928b98041f13a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55cf8f5b487dfb9ede36eb5c711adb5b.png)
③四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e689223dbda856ab0898b0010ffb86a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0868623e3c43201d1dd029b38d53b2ad.png)
【拓展应用】
(3) 如图3, 学校内有一块四边形的花圃
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0425abc0a23299964ede59c6a0732743.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b907d8eb3d3b70212e73e486398d89.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10b3086ec8aa1fe23d09d63a941a74ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ec9e2a600d4675d510c58b984027e33d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
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6 . 问题背景:如图,在正方形
中,边长为
.点
,
是边
,
上两点,且
,连接
,
,
与
相交于点
.
与
的关系,并说明理由;
(2)探索发现:若点
,
分别是
与
的中点,计算
的长;
(3)拓展提高:延长
至
,连接
,若
,请直接写出线段
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.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/5ea89499adc4ede5c30145f903d29699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
(2)探索发现:若点
![](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/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(3)拓展提高:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d5942a81561bf6617281126e97e5595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/892909e49156f7dcc0650fcd65243877.png)
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2024-04-23更新
|
297次组卷
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8卷引用:山东省烟台市招远市(五四制)2022-2023学年八年级下学期期中数学试题
山东省烟台市招远市(五四制)2022-2023学年八年级下学期期中数学试题山东省聊城市东昌府区韩集镇中学2023-2024学年下学期八年级期中模拟数学试题山东省德州市夏津县2022-2023学年八年级下学期期末数学试题广东省深圳市福田区耀华实验学校2023-2024学年九年级上学期月考数学试题甘肃省武威第三中学教研联片2023-2024学年八年级下学期数学第一次月考试题甘肃省武威市凉州区武威第四中学2023-2024学年八年级下学期3月月考数学试题湖北省孝感市重点校2023-2024学年八年级下学期月考数学试题(已下线)专题06梯形与中位线(8大题型)【好题汇编】-备战2023-2024学年八年级数学下学期期末真题分类汇编(上海专用)
7 . 已知菱形
,
.
,点E在边
上,点F在边
上,
,求证:
;
(2)如图2,
,点F在边
上,点E在边
上,
,过点F作
交
的延长线于点N,连接
,过点N作
交直线
于点H,求证:点F为
的中点;
(3)如图3,
,点E为边
的中点,点F在边
上,
,直接写出
的值______.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/597ce705d3a2fe04d29de9e81ec6250d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97cf714ffb3fd5917a76b191640b55fe.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/4a9eb456c912e01458c19b06e83a5c9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f805768a5ffaf8bdfa4bc3b680aafdc9.png)
(2)如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33ac762a2899a58faa0d3ab44f1281fa.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/345c91c9e67e6ee00a872078ab17f094.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8321f75ac5e40c4822c6965b8a47d42f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3009d7ff4719f6226722d06b14fae180.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e0ae50d5993a332b5cddb022eaa6f1e.png)
(3)如图3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97cf714ffb3fd5917a76b191640b55fe.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/88e5963f42d4195661bb0860b8d6c18d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52fa34269595fb910a27edb294e642e1.png)
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8 . 如图,正方形的边长为4,点E,F分别在
,
上,且
,
的延长线交
的延长线于点
,
的延长线交
的延长线于点
,连接
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/19/988c02c3-53e8-4d05-984e-a194c5f60427.png?resizew=505)
(1)填空:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e95777a5aa465745ef2011bd042b3216.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d890499dca9a8ef30addafe41d6e771.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/392cdb9d30684cce244bef94b8d861b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ff7942da6c3fc4005256fb1458557c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6706fe00b4e231e62d9ecbec567d526b.png)
(2)线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffc9d9b726f7b836910f4b48b971ea96.png)
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1bde5e96c203f387b2004b4abf5f839f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
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9 . 正方形
和等腰
共顶点D,
,将
绕点D逆时针旋转一周.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/13/265f75ed-f398-4b33-b9d0-619073344ea4.png?resizew=572)
(1)如图1,当点F与点C重合时,若
,求
的长;
(2)如图2,M为
中点,连接
,探究
的关系,并说明理由;
(3)如图3,在(2)条件下,连接
并延长交
于点Q,若
,在旋转过程中,
的最小值为 .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f73fbc2f33d3402cd96a776a376d9835.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e7de023b31141802459de509623e0a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/72cb97395ebc5ee1b212afb7a97b985c.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/13/265f75ed-f398-4b33-b9d0-619073344ea4.png?resizew=572)
(1)如图1,当点F与点C重合时,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/09d27bd71d79cb19eb554175e4ef0867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
(2)如图2,M为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf4939d216614a7434db5adfb2547569.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf4939d216614a7434db5adfb2547569.png)
(3)如图3,在(2)条件下,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15a424b50eaeafa6f302ffd95476cb86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0da89cfb8bdc95cd620fc7d528cc7942.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f53330c107f8245290a5a42c3d356acd.png)
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10 . 如图,在正方形
中,E、F分别是
,
的中点,
,
交于点G,连接
,下列结论:①
;②
;③
;④
,其中正确的结论是( )
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf857e03d4320c999d328fd657c2d412.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/783b84ed8d927b77491092f7d2ee2989.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4412d53360eca5a00f1d9925625ce86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/032a2b9d7d169436cdb842be77d2990b.png)
A.①② | B.①③ | C.①②④ | D.①②③ |
您最近一年使用:0次
2024-03-14更新
|
512次组卷
|
7卷引用:广东省汕头市澄海实验学校2022-2023学年九年级下学期期中数学试题
广东省汕头市澄海实验学校2022-2023学年九年级下学期期中数学试题广东省广州市越秀区广东实验中学2023-2024学年八年级下学期期中数学试题湖北省黄石市2023-2024学年八年级下学期期中数学试题河南省漯河市源汇区实验中学2023-2024学年八年级下学期第一次月考数学试题(已下线)第08讲 正方形(3大考点+6种题型+强化训练)-【帮课堂】2023-2024学年八年级数学下册同步学与练(苏科版)2024年广东省汕头市龙湖区汕头经济特区林百欣中学中考模拟数学试题(已下线)江苏省八年级下学期期中必刷压轴60题(27个考点专练)-2023-2024学年八年级数学下学期考试满分全攻略高频考点+重难点讲练与测试(苏科版)