名校
1 . 我们不妨约定:如果抛物线的顶点在直线
上,那么我们把这样的抛物线叫做“完美抛物线”,根据约定,解答下列问题:
(1)下列抛物线是“完美抛物线”的是______;
①
②
③![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcb247cbc25022c9ae5a79d1ba8b6b24.png)
【拓展应用】如图,已知“完美抛物线”
的顶点为
,将该抛物线沿直线
向上平移,点
平移到点
,两条“完美抛物线”相交于点
,设点
、点
的横坐标分别为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c6a7ae42ee41d2aa7977d541b91002.png)
(2)若
,求平移后的抛物线的解析式;
(3)在平移的过程中,若
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b9f0b9e53a83e68f5fec944f343119.png)
(1)下列抛物线是“完美抛物线”的是______;
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/344ccbf79da6ad7e3709d6fa72efb756.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d3ba3449c6cdbd05064a0179e2a1b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcb247cbc25022c9ae5a79d1ba8b6b24.png)
【拓展应用】如图,已知“完美抛物线”
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/761dae543f0d0bb45bd90a0f9506d7f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08b9f0b9e53a83e68f5fec944f343119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04c6a7ae42ee41d2aa7977d541b91002.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04725e51b4870658f74de79403e3898f.png)
(3)在平移的过程中,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2b7308135fda2e4b9b16457b6aa12df3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8fd1f0ace9ca0b79929e73af6c201c2e.png)
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2 . 【建立模型】(1)如图1,点B是线段
上的一点,
,
,
,垂足分别为C,B,D,
.求证:
;
的图象与y轴交于点A、与x轴交于点B,将线段
绕点B逆时针旋转
得到
、直线
交x轴于点D.
①点C的坐标为______;
②求直线
的解析式;
【拓展延伸】(3)如图3,抛物线
与x轴交于A,B两点(点A在点B的左侧),与y轴交于点C,已知点
,连接
,抛物线上是否存在点M,使得
,若存在,直接写出点M的横坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/615fc8790237a1b09af51d6bcad6b595.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf4dc4d7d30af1cdce660795e0fd7d7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5daa413c5a1b941452121c5d750a03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3d2d3643a9579f2c693ef86909441e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20d045ff0281b23804e15fa23ed40e13.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1d7df623642896d720d6956ed1f0ff6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
①点C的坐标为______;
②求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
【拓展延伸】(3)如图3,抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b73ec2e51e8bf082ec95b5ce8348de68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/608aac04088ef43a6e8f95815107c4aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb6ede9761b5b90f8dc137708e1ee90f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/114420d33bb74a43155a9ea85679527c.png)
您最近一年使用:0次
2024-04-11更新
|
349次组卷
|
4卷引用:2024年湖南省郴州市桂阳县蒙泉学校中考一模数学试题
名校
3 . 综合与实践:
问题情境:如图1,在正方形
中,点E是对角线
上一点,连接
,过点E分别作
的垂线,分别交直线
于点F,G.
和
的数量关系______.
(2)问题解决:如图2,在图1的条件下,将“正方形
”改为“矩形
”,其他条件不变.若
,求
的值;
(3)问题拓展:在(2)的条件下,当点E为
的中点时,请直接写出
的面积.
问题情境:如图1,在正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70708d5a85fd0904cc8a5c73353ce78f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0ced324a15510f9cf7518d8c277c6b50.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
(2)问题解决:如图2,在图1的条件下,将“正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee74b2491993ced6f30773dd39ede785.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0536bab866e0eeb95e1781af69c9aa4.png)
(3)问题拓展:在(2)的条件下,当点E为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3f3197779e26c71a1b633b279613969.png)
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2024-01-17更新
|
238次组卷
|
8卷引用:湖南省娄底市双峰县2023-2024学年九年级下学期月考数学试题
名校
4 . 【基础巩固】
(1)如图1,在
中,D为
上一点,连结
,E为
上一点,连结
,若
,求证:
.
【尝试应用】
(2)如图2,在平行四边形
中,对角线
交于点O,E为
上一点,连结
,若
,求
的长.
【拓展提升】
(3)如图3,在菱形
中,对角线
交于点O,E为
中点,F为
上一点,连结
,若
,
,求菱形
的边长.
(1)如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.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/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aadef7d71079f7953ea6eb3f9c44968.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cd7edc508c8fc8de42058bd2364d77a.png)
【尝试应用】
(2)如图2,在平行四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d0b0b7d07ac1aebab6b6f1e8c6acdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/828628c0876b45381c9a0edeb0fec236.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5168d2641ae584f6b922074320762f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/18ba99f3d174abf75fa5c9707f107145.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
【拓展提升】
(3)如图3,在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0d0b0b7d07ac1aebab6b6f1e8c6acdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1992f6d926022aa72f86d367da4bc8f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e511588db7f353f450c7a3a60135d699.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f1682d306c38087d9e6f7efb9cec596a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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5 . (1)问题初探:在直角三角形中,两直角边的长度之和是10,当两直角边的长分别是_______、_______时,直角三角形的面积最大;
(2)问题解决:如图①,在一个
的内部作一个矩形
,其中点A和点D分别在两直角边上,
在斜边上,
,
,矩形面积最大是多少?在解决这个问题时,有一位爱动脑筋的同学通过作辅助线进行了转化,如图①,过点D作
,所以
,又因为四边形
是矩形,所以
,
于是
,那么求矩形
的面积最大,就可以转化为求平行四边形
的面积最大,设平行四边形
的边
,平行四边形
的面积为
,请你按这个思路继续完成这问题;
(3)问题拓展:如图②,矩形
中,
,
,点E是
边上的动点(点E与A、D两点不重合),连接
、
,点F是
边上的动点,过F作
交
于G,求
面积最大值.
(2)问题解决:如图①,在一个
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80eff52574ec119d2f52fa165da837ec.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/4e70c1a2ee7155034a54dc615cc80813.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa0f561388f1c3046b9c53e955a3d41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/289be94b23fc4ebe26181ae626dc9906.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/541211a3ae17d1f4a337f82bbb26717d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c220eadc312101e2fb89dfe920f7b30d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4575d28fdf870383eff0114a9e18830b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ea9ef4a5db6c65316d90ed2a9ed038.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7398082f10073da5bfa37fd4c758af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7398082f10073da5bfa37fd4c758af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cd99679fe64a8fa1020b7fcb1db9a8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7398082f10073da5bfa37fd4c758af2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b997be2976c30e6d78037f53a62ee41.png)
(3)问题拓展:如图②,矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7df3b2901ad26337818f75e81448ebb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c545764505bb00578a870c5e39493a3e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a8ea7df6fb067a4ad237f77856a25c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36e5e61804ce550636a0354e0a78a22d.png)
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6 . [基础巩固](1)如图1,正方形
和正方形
,其中
,
,
三点共线,延长
交
于
,连接
.
(1)求证:
;
(2)不难证明:
,因此
的值为______;
[尝试应用](2)在(1)的条件下,如图1,若
,
,求正方形
的边长;
[拓展提高](3)如图2,正方形
和正方形
,
是
中点,连接
,
恰在
上,连接
,
,若
,求
的最小值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc6f18c9a80cdd89edce7bb775b51db8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e322e0c87479bba874db9ae9ba36b5.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/826c728050e3378921442ace20269ef6.png)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4577de33211bd254d2c54aa44b6b9fa6.png)
(2)不难证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5f57dbc0e3c0d1d21de6f31a0fc6cc6b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ea72d9e92de19fe73391107c84d54575.png)
[尝试应用](2)在(1)的条件下,如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/348fb71fbc47fd87e9ce011652ef4186.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6370d6c626bdabf1fc694501ee6c714f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc6f18c9a80cdd89edce7bb775b51db8.png)
[拓展提高](3)如图2,正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bc6f18c9a80cdd89edce7bb775b51db8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/16/ace79bf6-361a-4acf-b34f-89c3bf13bb07.jpg?resizew=308)
您最近一年使用:0次
2023-12-18更新
|
61次组卷
|
2卷引用:湖南省岳阳市通海路中学金凤桥校区2023-2024学年九年级上学期第三次月考数学试题
7 . 操作与研究:如图,
被平行于
的光线照射,
于D,
在投影面上.
的投影是______,线段
的投影是______.
(2)问题情景:如图1,
中,
,
,我们可以利用
与
相似证明
,这个结论我们称之为射影定理,请证明这个定理.
(3)拓展运用:如图2,正方形
的边长为15,点O是对角线
、
的交点,点E在
上,过点C作
,垂足为F,连接
;试利用射影定理证明
;
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b757f0c42ae5c9a2d6a4b19e5877b27.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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)问题情景:如图1,
![](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/8b757f0c42ae5c9a2d6a4b19e5877b27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9cf01992887d7c521e755623a82c44a1.png)
(3)拓展运用:如图2,正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/80125e48498ac9c548732e2af1b1758d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6cad4595d5352b2884568a59d8d766a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/633042a5f9af246d305120a96209e616.png)
您最近一年使用:0次
2023-11-10更新
|
168次组卷
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3卷引用:湖南省岳阳市汨罗市2023-2024学年九年级上学期期中数学试题
8 . 一次数学综合实践活动课上,小慧发现并证明了关于三角形角平分线的一个论证.如图1,已知
是
的角平分线,可证
.小慧的证明思路是:如图2,过点C作
,交
的延长线于点E,构造相似三角形来证明
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/24/ebae37d7-b4fd-47e4-bdd8-122651b3d5ac.png?resizew=509)
(1)尝试证明:请参照小慧提供的思路,利用图2证明
;
(2)应用拓展:如图3,在
中,
,D是边
上一点.连接
,将
沿
所在直线折叠,点C恰好落在边
上的E点处.
①若
,
,求
的长;
②若
,
,求
的长(用含k与
的代数式表示).
![](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/a02bad7fac070822f4c8b4f8b64bda72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b28971fe3e3f9d3849400e98b46fc3ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02bad7fac070822f4c8b4f8b64bda72.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/24/ebae37d7-b4fd-47e4-bdd8-122651b3d5ac.png?resizew=509)
(1)尝试证明:请参照小慧提供的思路,利用图2证明
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02bad7fac070822f4c8b4f8b64bda72.png)
(2)应用拓展:如图3,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c06154cae3bf7a8ce5a1e97a7380875.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/7ac451db3443cabb204f96c31fd4a02e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
①若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca036d049f5205cf04cb1b9c5cd03f97.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39a7a5e8311b6053ba227ee72a3cbdf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/524d4a0c1990fdca997bf894cfcd2cf0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e170f206fdbbd834aad7580c727e2cc6.png)
您最近一年使用:0次
名校
9 . 【问题背景】(1)如图1,已知正方形
和等腰
,
,O,H分别为
,
的中点.求证:
.
【变式应用】(2)如图2,已知菱形
和等边
,
,O,H分别为
,
的中点,连接
,
.求
与
的数量关系,并说明理由;
【拓展迁移】(3)如图3,已知
和
,
,
,O,H为
,
的中点,连接
,
,求
与
的数量关系(用k表示),并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c922cbd707131b52757c464804ca24f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a68f71476e365d4c354bd1f6963d1065.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c7bd8e6f1bee491929f3657f0e4a900f.png)
【变式应用】(2)如图2,已知菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f945a69cf7e8213e50622125cde652f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
【拓展迁移】(3)如图3,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb3673b5b0d16fe9af139bf24b0407d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a6ec5bc4c6dcb12222ce1371ef2469e.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/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/11/48155247-3fb6-4c41-9d32-cfe87c79307b.png?resizew=459)
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10 . 我们把两条中线互相垂直的三角形称为“中垂三角形”,例如图1,图2,图3中,
是
的中线,
,垂足为P.像
这样的三角形均为“中垂三角形”.设
,
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/14/c6e4d257-4cc1-46f0-92e5-46fef92dcd9c.png?resizew=133)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/14/ec91c45e-4a83-43f5-9e23-f2206a429733.png?resizew=211)
(1)[特例探索]
如图1,当
,
时,
______,
______;
如图2,当
,
时,
______,
______.
(2)[归纳证明]
请你观察(1)中的计算结果,猜想
,
,
三者之间的关系,用等式表示出来,并利用图3证明你发现的关系.
(3)[拓展应用]
利用(2)中的结论,解答下列问题:在边长为3的菱形
中,
为对角线
中点,
分别为线段
,
的中点,连接
并延长交于点
.
分别交
于点
,如图4所示,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/193a42f63312ac6a40d6e28ee93325aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/187ed13a7bd532bd39af5e5ad7493a2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8e781a2489271bfd1597cba1bb6f5887.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df81cda12d7601d58b1d9c7c180c4d66.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c884a45b56bc34d79273b067c1520b2.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/14/3b182c5b-78e2-4836-822d-4d0e746eef62.png?resizew=118)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/14/f15a7f3d-58c3-491d-93c4-300f8b96b7e0.png?resizew=138)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/14/c6e4d257-4cc1-46f0-92e5-46fef92dcd9c.png?resizew=133)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/14/ec91c45e-4a83-43f5-9e23-f2206a429733.png?resizew=211)
(1)[特例探索]
如图1,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eedf93beec809182ff56885c77cf18a9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad09921d9904335a83078262ce62a473.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380bbacf854e30e2e747fc286d2b9997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ccd4162c7d09f970cb77cadacdbe521.png)
如图2,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e58d4184f5b731327886e0586442de69.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60b97bb18e5ca34d22b5e827316a122a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380bbacf854e30e2e747fc286d2b9997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ccd4162c7d09f970cb77cadacdbe521.png)
(2)[归纳证明]
请你观察(1)中的计算结果,猜想
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c8cc0b4997cae4d8aec791a1d3923314.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a881309775c3b6a9f4ed408838666342.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/566f7614d51f0348db982a9440de8844.png)
(3)[拓展应用]
利用(2)中的结论,解答下列问题:在边长为3的菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a373959bb9026f8a09845c0b828bf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c3d2cba96f6f03520c0b3f6e4da03e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71b63d2504bd3ecce8c10560b142356f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ec07ca71c03f8c1d6d3a1e2efcdff02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36537f5405f1c2c35f5a03eb198724e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0f3ed5ea1cf0fa8f7c6be46cd5fa057.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d1e478ff979f8557e0f169eaf22f154.png)
您最近一年使用:0次
2022-11-13更新
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265次组卷
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5卷引用:湖南省新田县云梯学校2022-2023学年九年级上学期第二次阶段测试数学试题
湖南省新田县云梯学校2022-2023学年九年级上学期第二次阶段测试数学试题2019年江苏省徐州市铜山区九年级中考二模数学试题(已下线)2020届九年级《新题速递·数学》2月第01期(考点17)2023年江西省宜春市第八中学中考一模数学试题(已下线)2023年江西一模(几何综合)