1 . (1)问题发现:如图①,直线
,E是
与
之间的一点,连接
,可以发现
.
证明:过点E作
,
∵
(已知),
(辅助线的作法),
∴
( )
∴
.( )
∵
,
∴
(同理),
∴
(等式性质)
即
.
(2)拓展探究:如果点E运动到图②所示的位置,其他条件不变,猜想:
的关系,并说明理由.
(3)解决问题:如图③,
,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8a11029ca6b4b9e7f777af0280cf163c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/162ce08ead3fc7f199a7ca1616247abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a2470f3b13607a659b88dc6a85421d6.png)
证明:过点E作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25eff69a4a0dc7a7ab183843303d333.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee8ef58be8708144272538ee427fb92c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25eff69a4a0dc7a7ab183843303d333.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cdd721393a3d94cd3560678a50731792.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2525cb95d9803e61b25742fa255c5883.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b25eff69a4a0dc7a7ab183843303d333.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/afde09d82ae96f00c135732baee64776.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5614541ccf197a54b933cf229bb342ec.png)
即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a2470f3b13607a659b88dc6a85421d6.png)
(2)拓展探究:如果点E运动到图②所示的位置,其他条件不变,猜想:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/733d043b6e96440be8283315eb9f06f9.png)
(3)解决问题:如图③,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfb9551493a3827dd6128bb2a33cf7a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
您最近一年使用:0次
2 . 【思考尝试】
中,点
是
的中点,将正方形
沿
折叠,得到点
的对应点为
,延长
交线段
于点
,连接
.求
的度数.
【实践探究】
(2)小瑞受此问题启发,逆向思考并提出新的问题:如图②,正方形
的边长为6,点
,
分别在
,
上,连接
,
,
.若
,
,求
的长.
【拓展迁移】
(3)小波深入研究以上两个问题,发现并提出新的探究点:如图③,
是
的高,
,若
,
,求
的面积.
![](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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a9a7af8da37dc86b307285abf8e964d2.png)
【实践探究】
(2)小瑞受此问题启发,逆向思考并提出新的问题:如图②,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/a0ed1ec316bc54c37c4286c208f55667.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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/192155e6a3aade305b76b1eb7c75e30c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/214d99ce9f50286dcff41db85083c979.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
【拓展迁移】
(3)小波深入研究以上两个问题,发现并提出新的探究点:如图③,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f81e24376a13d648c2ed0dc73bc710e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a259656caff73f08304c865fe8f5205e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/430b77d8d2a99713b192dc729ddc2275.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
您最近一年使用:0次
3 . 【问题情境】已知,
,
平分
交
于点
.
(
)如图
,
,
,
,试判断
与
的位置关系,并说明理由;
【问题解决】
(
)如图
,
,若
,
时,求
的度数;
【问题拓展】
(
)如图
,若
,试说明
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0e53497af8899cb299d762f1a4f46a55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4598486f8c5ac28b1165b76e78105979.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
(
![](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/9f0b4b0020653b23cab99a99c658fe21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f544c72565338d679306e94e59a62249.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c7acc77d7705dd1b958cd6d739a4b01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
【问题解决】
(
![](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/483d4077054444cb15d496e87eb039d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fff774b4b0087a6f304ce930d359be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ba3963c52bb2b2a755f4a81013d25c3a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d335bf8cb2e6e5e50d0e03a14d1f48d4.png)
【问题拓展】
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0fff774b4b0087a6f304ce930d359be.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/87a18d4adfdbac0a902f31f9f222c0dc.png)
您最近一年使用:0次
名校
4 . 如图1,四边形
是平行四边形,点
是对角线
上一点,点
是
外一点,连接
、
和
,且
,
,
.
是菱形;
(2)【拓展延伸】如图2,连接
交
于点
,延长
交
于点
,求证:
;
(3)【实践应用】我校生态社在校内的蔬菜种植活动硕果累累,备受关注.如图3所示的一块正方形
种植区被
、
、
分割种植着不同植物,经测量得
,
.现学校决定延长
交
于点
,以
为边长,在该种植区的左边再开辟一块小正方形新区域种植更多蔬菜,求新区域的面积.
![](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/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc56c77464a17a1e97b568762a3e2c6.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/ac679f877d3e3b3acef1f1f8e3654b4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad4c5d9a889842bb78bcf12f0909c4c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4bbe928cbfa7ccd2d93a08f7a92861.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)【拓展延伸】如图2,连接
![](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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a74fa1d77910529e4eda3b2d70f40a1.png)
(3)【实践应用】我校生态社在校内的蔬菜种植活动硕果累累,备受关注.如图3所示的一块正方形
![](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/5634a208294350f8ec151cd19b6d3278.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e5f736b1195fef1d2d300168a795f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a3df7035b26938612586bbd963914fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34d795825a51d2a1954018548255f91c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4505508b3e36db64a207dcdaf8eb22dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
您最近一年使用:0次
5 . 【知识再现】如图1,四边形
是正方形,点M,N分别在边
,
上,连接
,
,
,
,延长
至点G,使
,连接
.根据全等三角形的知识,我们可以证明
.
,垂足为H,猜想
与
有怎样的数量关系,并给出证明.
(2)【知识应用】如图2,已知
,
于点D,且
,
,则
的长为__________.
(3)【知识拓展】如图3,四边形
是正方形,E是边
的中点,F是边
上一点,
,
,求
的长.
![](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/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4794cf7c57bdd4825d9e6615e2527a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d74d7ac8bc3f82ce121c0065a928a59.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d8c24906872c98c181f198ee0ab6e48.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41e9c029e8c099fecbf785a18559196.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)【知识应用】如图2,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3f81e24376a13d648c2ed0dc73bc710e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/55c24a968c73e960698a572ab01e3698.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267ace52b64e1e7dfc5211e033255b7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(3)【知识拓展】如图3,四边形
![](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/a29b1ba43a08d5ce726948c75950217a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44a1f73190215b1c9d45a3dd9e6b66ce.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
您最近一年使用:0次
6 . 根据项目素材,探索解决问题.
项目主题 | 如何剪出直角三角形的完美线? |
项目背景 | 新课标(2022版)要求学校教育要坚持“立德树人”,实施“跨学科学习、项目式学习”.在学习完特殊的平行四边形后,某校组织了该次项目式学习. |
项目素材 | 在直角三角形中,过直角顶点剪一刀,剪痕将直角分成两个锐角,若这两个锐角分别等于此直角三角形中的另外两个内角,则称这条剪痕为直角三角形的“完美线”. |
问题解决 | |
项目一 操作 | 如图,有一张直角三角形纸片,![]() ![]() ![]()
|
项目二 探究 | 如图,在直角三角形纸片中,![]() ![]() ![]() ![]()
|
项目三 拓展 | 在![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
您最近一年使用:0次
名校
7 . (1)【问题提出】
如图1,在
中,
,
,点D为边
上一点,过D作
于E点,连接
,F为
的中点,连接
,
,
,则
的形状是______
(2)【问题探究】
如图2,将图1中的
绕点B按逆时针方向旋转,使点D落在
边上,试判断
,
,
的数量关系,并说明理由;
(3)【拓展延伸】
若
,
,将
绕点B按逆时针方向旋转,当点D在线段
上时,直接写出线段
的长______(用含m的式子表示).
如图1,在
![](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/ab926d89b65f26c12e3da73ef1e5cf68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c38dfd14dde969702dff97ef2270f9.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/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ff39c7aa648afd1080206c8080ff79e.png)
(2)【问题探究】
如图2,将图1中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/321370fe42bc1216902ea19fbd2a5979.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(3)【拓展延伸】
若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc2d9d9adad09281ca67132bdce523fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3704a0e07f4c91f2b5047cd20cae1fd5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/321370fe42bc1216902ea19fbd2a5979.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
您最近一年使用:0次
2024-06-09更新
|
265次组卷
|
4卷引用:2024年福建省厦门市湖里中学中考二模数学试题
名校
8 . 如图
,将一副三角板的直角顶点
叠放在一起.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/31/691dd21a-05e8-41aa-acee-063edaa99de1.png?resizew=435)
(1)观察分析∶若
则![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41db498f025052df2dbb1e50ab672675.png)
_______ ,若
则
______________ ;
(2)猜想探究∶如图
,若将两个同样的三角尺,
锐角的顶点
重合在一起,请你猜想
与
有何关系,请说明理由;
(3)拓展应用∶如图
,如果把任意两个锐角
的顶点
重合在一起,已知
,
(
、
都是锐角),请你直接写出
与
的关系.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/31/691dd21a-05e8-41aa-acee-063edaa99de1.png?resizew=435)
(1)观察分析∶若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a558cf48126f668a4683dda175c042a3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41db498f025052df2dbb1e50ab672675.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f085f746975ead232fd362d5283c69c1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9454959fd953ac3d66f2a325a553f432.png)
(2)猜想探究∶如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be6a6301878fed2a01413020b27310a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4189a0821a0ffab9dc171ecd279ba442.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/35803969509655f72b4118503ae1c966.png)
(3)拓展应用∶如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2b660a997b88f192845b2166ef67e36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eec6ce73c6786d50028addff089bbb64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/654287b1c90c3d14d22b5c36cc8f930c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b5858ee1ce52b251816757257a11c29.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/42e7b8a59f8d66176463f3d71f882a26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d22df2977de56cc69be0c1e847653d7a.png)
您最近一年使用:0次
2024-02-27更新
|
109次组卷
|
2卷引用:福建省福州第七中学2023-2024学年七年级上学期期末数学试题
9 . 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
翻折到
的位置;
第3步:延长
交
于点H,则点H为
边的三等分点.
第1步:如图2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:再将正方形纸片对折,使点B与点D重合,再展开铺平,折痕为
,沿
翻折得折痕
交
于点G;
第3步:过点G折叠正方形纸片
,使折痕
.
【过程思考】
(1)“乘风”小组的证明过程中,三个空的所填的内容分别是①:______,②:______,③:______;
(2)结合“破浪”小组操作过程,判断点M是否为
边的三等分点,并证明你的结论;
【拓展提升】
如图3,在菱形
中,
,
,E是
上的一个三等分点,记点D关于
的对称点为
,射线
与菱形
的边交于点F,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/03902478df1a55bc99703210bccab910.png)
证明过程如下:连接![]() ∵正方形 ![]() ![]() ∴ ![]() 又∵ ![]() ∴ ![]() ∴ ![]() 由题意可知E是 ![]() ![]() ![]() 则 ![]() 在 ![]() 解得: ![]() ![]() |
第1步:如图2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
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
第2步:再将正方形纸片对折,使点B与点D重合,再展开铺平,折痕为
![](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)
第3步:过点G折叠正方形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ff3c1ea812dbddf9c5752e45e3713a7.png)
【过程思考】
(1)“乘风”小组的证明过程中,三个空的所填的内容分别是①:______,②:______,③:______;
(2)结合“破浪”小组操作过程,判断点M是否为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
【拓展提升】
如图3,在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f08273d339dc5ddbb89aa67bb8205e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/454328a8e75953fdb0835ce80d9566e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d226aef207ce71a381d6f63801cc9d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/557f8ff92d79a9d464ff13de17f3eae7.png)
您最近一年使用:0次
2024-02-24更新
|
287次组卷
|
2卷引用:福建省泉州实验中学2023-2024学年九年级下学期期中数学试题
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)
您最近一年使用:0次