1 . 小亮同学喜欢研究数学问题.他在一本资料中看到一个新的数学概念“对角线互相垂直且相等的四边形叫做垂等四边形”,并对垂等四边形进行了研究.具体内容如下:
【理解应用】
(1)如图1,在平面直角坐标系
中,已知四边形
是垂等四边形,点
的坐标为
,点
的坐标为
,求点
的坐标;
(2)如图2,正方形
的边长为
,点
在边
上,点
在边
上,点
在边
上,点
在边
上,若四边形满足
,请直接写出四边形
面积
的取值范围;
(3)如图3,已知抛物线
与
轴交于
、
两点,点
在点
的左侧,
、
两点在该抛物线上.若以
、
、
、
为顶点的四边形是垂等四边形且
.设点
的横坐标为
,点
的横坐标为
,且
,求
的值.
【理解应用】
(1)如图1,在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3241d7fedd89d85711acd7a2635298af.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cfa2a6d9749a619edf80bad8b3e4962.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fed700b7a3ab2ee7fbae12507033af9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
(2)如图2,正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9016de3d17a389fb5d0952f616c19e0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/611f100dcfa7803db6eb233e2e7f2dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
(3)如图3,已知抛物线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3236d825b994ee9c28e5d5479a57b8ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81dea63b8ce3e51adf66cf7b9982a248.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/966a7257fd3af3a4049844773c611e26.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7091d529281abff275ef19b9197445a7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
您最近一年使用:0次
2 . 【课例改编】
数学课上,张老师根据数学课本习题改编了一个题目:如图,
是
的高,
,若
,求
的长.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/69570adb-f3d8-4a72-b25a-459eb3f61801.png?resizew=179)
小明同学的想法是利用构造全等三角形来解决:将
沿
折叠,如图1,则点
刚好落在
边上的点
处.……
(1)结合小明同学的想法,请直接写出:
_____.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/f6c396b9-2ba8-45ef-b4d1-c5a17d2df8f4.png?resizew=177)
【改编拓展】
张老师继续启发同学们改编此题,得到下列试题,请同学们解答:
(2)如图2,
为
的外角
的平分线,交
的延长线于点
,则线段
有什么数量关系?请写出你的猜想并证明.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/bf43183d-9280-44c4-8d08-f66bf4ceb32d.png?resizew=170)
【模型应用】
根据上面探究构造全等模型的规律,请解答:
(3)如图3,在四边形
中,
平分
,求
的长.
数学课上,张老师根据数学课本习题改编了一个题目:如图,
![](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/1c9af7ab732d431dd78e84db9586d3cc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1fb8f8236e1c649fac70e7b08343145.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/69570adb-f3d8-4a72-b25a-459eb3f61801.png?resizew=179)
小明同学的想法是利用构造全等三角形来解决:将
![](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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
(1)结合小明同学的想法,请直接写出:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16ca90e8a784f990c4097eec9219908d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/f6c396b9-2ba8-45ef-b4d1-c5a17d2df8f4.png?resizew=177)
【改编拓展】
张老师继续启发同学们改编此题,得到下列试题,请同学们解答:
(2)如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd13d5c5788196f332fe53f85a0ad86e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7db0641078052bf1af17b4c5a01b9ccc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ec863999b1974aabb6dc67841765161.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/bf43183d-9280-44c4-8d08-f66bf4ceb32d.png?resizew=170)
【模型应用】
根据上面探究构造全等模型的规律,请解答:
(3)如图3,在四边形
![](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/4ecf6fbd7efd5867cbff635bea788a1f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/29/12641f29-772f-4421-b354-5e58f726c3b8.png?resizew=168)
您最近一年使用:0次
2024-01-27更新
|
236次组卷
|
2卷引用:2024辽宁省大连市中考一模考前数学调研题
名校
3 . 【阅读材料】小高同学发现这样一个规律:两个顶角相等的等腰三角形,如果具有公共的顶点的顶点,并把它们的底角顶点连接起来则形成一组全等的三角形,小高把具有这个规律的图形称为“手拉手”图形.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/12/0c4228fb-3419-4c8c-aeba-19e77c73beb6.png?resizew=576)
【材料理解】
(1)如图1,在“手拉手”图形中,小高发现若
,
,
,则
,请证明小高的发现.
【深入探究】
(2)如图2,
,
,
,试探索线段
,
,
之间满足的等量关系,并证明结论;
【延伸应用】
(3)①如图3,在四边形
中,
,
,
,
与
的数量关系为:________(直接写出答案,不需要说明理由);
②如图4,在四边形
中,
,若
,
,则
的长为________(直接写出答案,不需要说明理由).
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/12/0c4228fb-3419-4c8c-aeba-19e77c73beb6.png?resizew=576)
【材料理解】
(1)如图1,在“手拉手”图形中,小高发现若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8abfdaab699cfe51bb9678110ad6aee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d212c1709b8e72a055cf1b5381ef64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52273305805769a438772342b53c289e.png)
【深入探究】
(2)如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26813466e2ee49a493881a4384fc8748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2d212c1709b8e72a055cf1b5381ef64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
【延伸应用】
(3)①如图3,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532aece6cfd67e2a97977eed978dbf2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3d2d3643a9579f2c693ef86909441e2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2edd7dbb895fa95a022eb02bbe89d62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3818a2c9919d358b4c3713396093822b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eb3de20ba8bda63c650ef571be155702.png)
②如图4,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2894a6ff5b198184fffcaa1d03a33227.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32f01c4faacedfe56f5127d6c0cc63cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/037b342a682cbd4241855a243da3c016.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
您最近一年使用:0次
2023-01-09更新
|
497次组卷
|
3卷引用:山东省淄博市张店区张店区第九中学2022-2023学年九年级下学期期中数学试题
4 . 某数学兴趣小组对一个数学问题的探究过程如下,请仔细阅读,并解答相应问题.
【问题】如图,
中,
,
cm,D为
边上一个动点,连接
,过点C作
,垂足为点E,F为线段
上一点,且
,过点F作
交直线
于点G,判断线段
,
,
的数量关系.
【观察】数学兴趣小组的同学观察到线段
,
,
的长度随
的长度变化而变化,但他们并没有发现明显规律.
【实验】他们借助电脑软件根据点D在
上的不同位置,测量线段
,
,
,
的长度,得到下表的几组对应值.
请根据以上信息,完成下列问题.
【猜想】(1)线段
,
,
的数量关系为____.
【证明】(2)请证明上述猜想.
【拓展】(3)上述问题中,若D为射线
上的一个动点,F为射线
上的一个动点,其他条件不变,当
cm时,直接写出
的长.
【问题】如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed10df4140819d5451773a45de66201b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f13e584349ba3775c49424fe646d8460.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/497846628a41a9bc750a645e045afb47.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1642eec556eb252de9c1ab7bb5ca90b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88bbc115577cd400ad9b83b0102440c0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1a946f9010d77f54d566e338ee13b76a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9abaeba15f3abdd877bc701af52c5cd9.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/e0e57a13c665af88f326c9890072bf73.png)
【观察】数学兴趣小组的同学观察到线段
![](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/e0e57a13c665af88f326c9890072bf73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
【实验】他们借助电脑软件根据点D在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.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/e0e57a13c665af88f326c9890072bf73.png)
![]() | 1.00 | 2.00 | 3.00 | 4.00 | 5.00 |
![]() | 0.98 | 1.85 | 2.57 | 3.12 | 3.54 |
![]() | 0.20 | 0.74 | 1.54 | 2.50 | 3.54 |
![]() | 0.78 | 1.11 | 1.03 | 0.62 | 0 |
【猜想】(1)线段
![](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/e0e57a13c665af88f326c9890072bf73.png)
【证明】(2)请证明上述猜想.
【拓展】(3)上述问题中,若D为射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1642eec556eb252de9c1ab7bb5ca90b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a37838f9e82d7fcebfd8ec0f192dfab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/3/2a701aea-d86a-45eb-bd82-7bfcb781ea20.png?resizew=280)
您最近一年使用:0次
5 . 在
中,
,点D是边
上一点,过点D作
于点E,点F是边
的延长线上一点,且
,连接
,设
两点间的距离为
两点间的距离为
两点间的距离为
.
根据学习函数的经验,我们对因变量随自变量x的变化而变化的规律来进行探究.
(1)列表:下表的已知数据是根据
两点间的距离x进行取点、画图、测量,分别得到的几组对应值:
请你通过计算补全表格:
______;
(2)描点、连线:在平面直角坐标系
中,描出表中各组数值所对应的点
,并画出函数
关于x的图象、函数
关于x的图象;
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/29/5f447c97-dc4e-4899-b3b5-6e0849f1ca43.png?resizew=374)
(3)探究性质:请写出一条
关于x的函数的性质:_________;
(4)解决问题:当
是等腰三角形时,x大约是______
(结果保留一位小数).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8f88798ec42a58dccd212586382b23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/184874f89463068ebc7c9baf851ee206.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68fbbd17c89f03dbb61cd6ffdb9a0344.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78bc939ef2208ac7c9c07b2cca587abd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dca8a2ccce217c5cef108c19d92b1326.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa69fd8445d01c98634c2e885b47d5a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d8259fb02e85ad9f11c9e750ac4a903.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dc7001bf6c904ee2528e22c3de26241.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3085f96c87b66a354ff38ce3b64dcfbd.png)
根据学习函数的经验,我们对因变量随自变量x的变化而变化的规律来进行探究.
(1)列表:下表的已知数据是根据
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5aa69fd8445d01c98634c2e885b47d5a.png)
![]() | 0 | 0.25 | 0.5 | 0.75 | 1 | 1.25 | 1.5 | 1.75 | 2 | 2.25 | 2.5 | 2.75 | 3 |
![]() | 4 | 4.01 | 4.02 | 4.03 | 4.05 | 4.10 | 4.16 | 4.24 | 4.33 | 4.45 | 4.61 | 4.78 | a |
![]() | 3 | 3.07 | 3.17 | 3.27 | 3.37 | 3.54 | 3.70 | 3.88 | 4.08 | 4.29 | 4.55 | 4.82 | 5.11 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/380bbacf854e30e2e747fc286d2b9997.png)
(2)描点、连线:在平面直角坐标系
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63182f620ffd33adcef47bcac31b59a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/524abb4156c79c2c1c69ce3d2d090ff0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c6c18e0a00b092e3119f03bbb85ec3b.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/7/29/5f447c97-dc4e-4899-b3b5-6e0849f1ca43.png?resizew=374)
(3)探究性质:请写出一条
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/524abb4156c79c2c1c69ce3d2d090ff0.png)
(4)解决问题:当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c105d6ba18fbb0581fb982175e2eac9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efa9fbcfb9595e2f031aa691db4564b.png)
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解题方法
6 . [探索规律]
如图①,在△ABC中,点D、E、F分别在AB、 BC、 AC上,且DF//BC,EF//AB.设△ADF的边DF上的高为h1,△EFC的边CE上的高为h2.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/22/c71a5258-4bff-47fc-905e-b3ea84f955df.png?resizew=278)
(1)若△ADF、△EFC的面积分别为4和1,则
=______;
(2)某校数学兴趣小组的同学对△ADF、△EFC、四边形BDEF的面积关系进行了研究设△ADF、△EFC、四边形BDEF的面积分别为S1、 S2、S, EC的长为a,则S2=______ (用含a和h2的式子表示);S1=_____ (用含a、h1和h2的式子表示);S=______(用含a、h1的式子表示);从而得出S=2
.
[解决问题]
(3)如图②,在△ABC中,点D、E分别在AB、AC上,点F、G在BC上,且DE//BC,DF//EG.若△ADE、△DBF.△EGC的面积分别为2、3、 5,求△ABC的面积.
如图①,在△ABC中,点D、E、F分别在AB、 BC、 AC上,且DF//BC,EF//AB.设△ADF的边DF上的高为h1,△EFC的边CE上的高为h2.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/22/c71a5258-4bff-47fc-905e-b3ea84f955df.png?resizew=278)
(1)若△ADF、△EFC的面积分别为4和1,则
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0bc9e7badfafa2bfd9ece72da1ac71a.png)
(2)某校数学兴趣小组的同学对△ADF、△EFC、四边形BDEF的面积关系进行了研究设△ADF、△EFC、四边形BDEF的面积分别为S1、 S2、S, EC的长为a,则S2=______ (用含a和h2的式子表示);S1=_____ (用含a、h1和h2的式子表示);S=______(用含a、h1的式子表示);从而得出S=2
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/66950fa90903fa630a037586c57be29b.png)
[解决问题]
(3)如图②,在△ABC中,点D、E分别在AB、AC上,点F、G在BC上,且DE//BC,DF//EG.若△ADE、△DBF.△EGC的面积分别为2、3、 5,求△ABC的面积.
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2020-07-18更新
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7 . 【背景知识】研究平面直角坐标系,我们可以发现一条重要的规律:若平面直角坐标系上有两个不同的点
、
,则线段AB的中点坐标可以表示为![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c8c26ef38b4e89509e9063135750770.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/17/3d93f7d1-b200-46a5-a4d0-a6d4ab2a7627.png?resizew=470)
【简单应用】如图1,直线AB与y轴交于点
,与x轴交于点
,过原点O的直线L将
分成面积相等的两部分,请求出直线L的解析式;
【探究升级】小明发现“若四边形一条对角线平分四边形的面积,则这条对角线必经过另一条对角线的中点”
如图2,在四边形ABCD中,对角线AC、BD相交于点O,
试说明
;
【综合运用】如图3,在平面直角坐标系中
,
,
,若OC恰好平分四边形OACB的面积,求点C的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fe47a813432a63b7f4afd94a4fa9107.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc1dbd7e2f60f0dcd559209d35bec6e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c8c26ef38b4e89509e9063135750770.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/17/3d93f7d1-b200-46a5-a4d0-a6d4ab2a7627.png?resizew=470)
【简单应用】如图1,直线AB与y轴交于点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60010cba80d14fea830ee1f03c6dad6f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d552ee3c32a8976ff5767daaafac770c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12cfce6bedc9e8962a4a8b9afce4d4a9.png)
【探究升级】小明发现“若四边形一条对角线平分四边形的面积,则这条对角线必经过另一条对角线的中点”
如图2,在四边形ABCD中,对角线AC、BD相交于点O,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4252d7a512f43ed2539b0e04afdffdd2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4738132e10d0dfad2e3d82e18a1765e6.png)
【综合运用】如图3,在平面直角坐标系中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/38b73e6777246f78353fc101dd1bc619.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/81e1402fb671b426875a5af5e0cd467b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2a5032c324d66b4b38fa7f05013f205.png)
您最近一年使用:0次
2019-04-10更新
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8卷引用:山东省济宁市徽山县璎轩实验中学2022-2023学年九年级下学期第一次月考数学试题