问题情境:“综合与实践”课上,老师提出如下问题:将图1中的矩形纸片沿对角线剪开,得到两个全等的三角形纸片,表示为
和
,其中
.将
和
按图2所示方式摆放,其中点
与点
重合(标记为点
).当
时,延长
交
于点
.试判断四边形
的形状,并说明理由.
(2)深入探究:老师将图2中的
绕点
逆时针方向旋转,使点
落在
内部,并让同学们提出新的问题.
时,过点
作
交
的延长线于点
与
交于点
.试猜想线段
和
的数量关系,并加以证明.请你解答此问题;
时,过点
作
于点
,若
,求
的长.请你思考此问题,直接写出结果.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fd09f591e95fd40e8ecfe6020f9f46.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd3578471c9b37dee9cf3e6fd706d2e4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3fd09f591e95fd40e8ecfe6020f9f46.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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f10327213a940d94a796ba0bb46d42a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/693cd6179b2a92f03153ce12a0e86b95.png)
(2)深入探究:老师将图2中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abe4bdc5d9e833b23a1b916c06fc1a35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd9a666965db844e9000495b94124b60.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd7e284eac4b90bfb327de768a7beef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/889696ca1742000a208a7c2f40c3768e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61ba4c2b1162042e43e5081c7a2d3fda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b23393d953a5bcb37b595caba358f34d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8dd9ffec335ea9c3adcacd8790afe783.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
2023·山西·中考真题 查看更多[24]
2023年山西省中考数学真题(已下线)专题30 新定义与阅读理解创新型问题(共31题)-学易金卷:2023年中考数学真题分项汇编(全国通用)(已下线)专题17 几何压轴题-学易金卷:5年(2019-2023)中考1年模拟数学真题分项汇编(全国通用)(已下线)2023年山西省中考数学真题变式题19-21题广东省深圳市深圳实验学校初中部2023-2024学年九年级上学期月考数学试题(已下线)XDRzkgssxtzxl954福建省三明市三元区2023-2024学年九年级上学期期中数学试题浙江省宁波市海曙区高桥中学等四校2023-2024学年九年级上学期期中联考数学试题辽宁省沈阳市皇姑区第四十三中学2023-2024学年九年级上学期期中数学试题辽宁省阜新市彰武县2023-2024学年九年级上学期期末数学试题河南省周口市商水县2023-2024学年九年级上学期期末数学试题28.1锐角三角函数(已下线)第5讲 探究题江苏省苏州市 吴中区西安交通大学苏州附属中学2023-2024学年九年级下学期3月月考数学试题江苏省泰州市姜堰区第四中学2023-2024学年九年级下学期3月月考数学考试试题山东省德州市乐陵市致远实验学校2023-2024学年下学期九年级数学月考题湖北省荆门市京山市2023-2024学年九年级下学期期中数学试题广东省珠海市华发容闳学校数学2023-2024学年九年级下学期月考数学试题2024年江西省抚州市南城县中考一模数学试题(已下线)专题11 四边形压轴题综合-2024年中考数学二轮热点题型归纳与变式演练(全国通用)山东省威海市荣成市荣成市实验中学2023-2024学年九年级下学期期中数学试题2024年山东省临沂市蒙阴县中考三模数学试题2024年河南省周口市沈丘县2校中考二模联考数学试题2024年山东省临沂市沂水县中考数学二模试题
更新时间:2023-06-23 08:57:03
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相似题推荐
解答题-证明题
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名校
【推荐1】如图,矩形
,点D在
上,将矩形
沿
折叠,点A的对应点C落在
上,
(1)如图1,求证:四边形
为正方形.
(2)如图2,四边形
中,点G在
上,点H在
上,
,连接
,求证:
.
(3)如图3,在(2)的条件下,连接
分别交
,
于点T、K,连接
,若
,
的面积为
时,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/6/6/17911f2d-79a6-4df3-ae84-792f0a16e7c2.png?resizew=531)
(1)如图1,求证:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)如图2,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4978985f24d9a98892cb291b30c841c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e42887d9bf31c1dd99f13c39e63c9ab9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5153b36b920d2be0cad78da0971b0218.png)
(3)如图3,在(2)的条件下,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06e322e0c87479bba874db9ae9ba36b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd319b129adce519612311ec4a3d82d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2afbd137b0f5f380c8abf1922f189bf3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8749c593f55086b418286727dfa0b53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b7f27ebcef70a3ebbbe8d2e53ea0896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1b6fb0185d4a13fae9d8dab8b4e18a68.png)
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【推荐2】我国古代建筑屋顶大部分属于坡屋顶的范畴.与平屋顶相比,其优点是排水迅速、不易积水,所以一般不会形成渗漏并影响下部结构.各种坡屋顶类型早在秦汉时期就已基本形成,到宋代更为完备.可以将房脊抽象成数学问题.如图,
分别与
相切于点
,连接
.连接
,交
于点
,交
于点
.
延长交
于点
,
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/10/0e687959-2298-4df0-969f-6e7e8b597955.png?resizew=493)
(1)若
,①连接
,判断四边形
的形状,并说明理由.
②若
的半径为
,直接写出劣弧
的长为______.
(2)若
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/790ef3382b1c731f2885eecfd92c2a86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39acab3cfb59bfc9591371721ab01d93.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef49a3ca580a144cc65a609c167facc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.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/ef49a3ca580a144cc65a609c167facc1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/5/10/0e687959-2298-4df0-969f-6e7e8b597955.png?resizew=493)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/722e40a19572920f0e5a10ec2f06a05b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9ba6d3a4b110f65a8d203aeadd03ecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c39a4d3936f26f94f4e85a1be4eefd66.png)
②若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fbe9f7abf7bcf4e1aa2579cd191d7761.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac1dda8823737f7d33aa21c8f35fd334.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
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【推荐3】解题与遐想.
王小明:
这道题算出来面积刚好是
,太凑巧了吧.刚好是
,有种白算的感觉![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
赵丽华:
我把
和
换成
、
再算一遍,
的面积总是
!确实非常神奇了![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
数学刘老师:
大家想一想,既然结果如此简单到极致,不计算能不能得到呢?比如,拼图?
霍佳:
刘老师,我在想另一个东西,这个图能不能尺规画出来啊感觉图都定了.我怎么想不出来呢?
拼图演绎
(1)将
分割放入矩形中(如左图),现在为了通过拼图能直接“看”出“
”,请在右图中画出拼图后的
个直角三角形甲、乙、丙、丁的位置,作必要标注.
(2)尺规作图:如图,点
在线段
上,以
为斜边求作一个
,使它的内切圆与斜边
相切于点
.
(保留作图的痕迹,写出必要的文字说明)
如图,![]() ![]() ![]() ![]() ![]() 求: ![]() |
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/26/51cae68d-6ee2-4b91-b4c4-a4f3985896db.png?resizew=162)
王小明:
这道题算出来面积刚好是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b7f27ebcef70a3ebbbe8d2e53ea0896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26a03dbc48c0a73bbf84851eec6d7784.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
赵丽华:
我把
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6a24198bd04c29321ae5dc5a28fe421.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb74ca8fc86ddef279e33f31c1fedda.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
数学刘老师:
大家想一想,既然结果如此简单到极致,不计算能不能得到呢?比如,拼图?
霍佳:
刘老师,我在想另一个东西,这个图能不能尺规画出来啊感觉图都定了.我怎么想不出来呢?
拼图演绎
(1)将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b7f27ebcef70a3ebbbe8d2e53ea0896.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/26/ef287103-401a-4ee3-a610-7355e8fc4276.png?resizew=411)
(2)尺规作图:如图,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
(保留作图的痕迹,写出必要的文字说明)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/26/2a32e14e-40e7-41bc-9dc6-137f7566bc9f.png?resizew=350)
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【推荐1】【问题情境】
在一次数学兴趣小组活动中,小昕同学将一大一小两个三角板按照如图1所示的方式摆放.其中
.
【问题探究】
小昕同学将三角板
绕点B按顺时针方向旋转.
上时,延长
交
于点F,求
的长.
(2)若点C、E、D在同一条直线上,求点D到直线
的距离.
(3)连接
,取
的中点G,三角板
由初始位置(图1),旋转到点C、B、D首次在同一条直线上(如图3),求点G所经过的路径长.
(4)如图4,G为
的中点,则在旋转过程中,点G到直线
的距离的最大值是______.
在一次数学兴趣小组活动中,小昕同学将一大一小两个三角板按照如图1所示的方式摆放.其中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab06e284377ddefeaa91173e9f1543c.png)
【问题探究】
小昕同学将三角板
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ab8a10e675354fa0c6e7da3d06b999d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(2)若点C、E、D在同一条直线上,求点D到直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(3)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4ab8a10e675354fa0c6e7da3d06b999d.png)
(4)如图4,G为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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【推荐2】已知四边形ABCD中∶
,
的延长线与BC的延长线交于点F,延长BE交
于点H,连接
,
,求证:△BCE≌△DCF;
(2)如图②,四边形ABCD为矩形,点E为CD边上一点,连接BE,作点D关于BE的对称点
,
的延长线与BC的延长线交于点F,连接
,
,
.如果
,AB=2,BC=3,求
的长;
(3)如图③,已知四边形ABCD为菱形,
,AC=8,点F为线段BD上一动点,将线段AD绕点A按顺时针方向旋转,当点D旋转后的对应点E落在菱形的边上时(顶点除外),如果DF=EF,请直接写出此时OF的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b25f3ea33cc08b1e2a0d9c3a9dccaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b25f3ea33cc08b1e2a0d9c3a9dccaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32b435d7fc33860ae191f9111d880b40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f611f5bb08a66cae8fe411e59a1c08e.png)
(2)如图②,四边形ABCD为矩形,点E为CD边上一点,连接BE,作点D关于BE的对称点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/82b25f3ea33cc08b1e2a0d9c3a9dccaf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32b435d7fc33860ae191f9111d880b40.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ba89e83329983cfadbfcdda151aaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f611f5bb08a66cae8fe411e59a1c08e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca359f630707bfe5196b6b817c79585.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e8ba89e83329983cfadbfcdda151aaa3.png)
(3)如图③,已知四边形ABCD为菱形,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a91146605aff8a321203a5907143cd2.png)
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【推荐1】如图,在平面直角坐标系中,已知抛物线
与x轴交于
,
两点,与y轴交于点C,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/3/0a026506-3f71-4a79-999f-731addd65920.png?resizew=545)
(1)求该抛物线的解析式;
(2)点P为直线
上方的抛物线上一点,过点P作y轴的垂线交线段
于M,过点P作x轴的垂线交线段
于N,求
的周长的最大值.
(3)若点N为抛物线对称轴上一点,抛物线上是否存在点M,使得以B,C,M,N为顶点的四边形是平行四边形?若存在,请求出所有满足条件的点M的坐标;若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8dd4534f6b65873203976e56d84e5b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0929421a6188c3122442866b0b85a5e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ef03f452410ab19c6246567c427178.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/4/3/0a026506-3f71-4a79-999f-731addd65920.png?resizew=545)
(1)求该抛物线的解析式;
(2)点P为直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1ed4c4e8edbd179f3fc38a6653f18c1.png)
(3)若点N为抛物线对称轴上一点,抛物线上是否存在点M,使得以B,C,M,N为顶点的四边形是平行四边形?若存在,请求出所有满足条件的点M的坐标;若不存在,请说明理由.
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【推荐2】综合与实践
在一次综合实践活动课上,小明将锐角三角形纸片沿过点
的直线折叠,使边
落在
边上,展开纸片,折痕为
,如图(
).展开纸片,再次折叠,使点
与点
重合,折痕为
,展开纸片,连接
,
,如图(
).
(1)【探索发现】小明发现折痕
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ffb98f1e3c1317c0db403d3af04bdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1ffb98f1e3c1317c0db403d3af04bdc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a02bad7fac070822f4c8b4f8b64bda72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
(2)【拓展应用】有一张三角形纸片
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/787ac5e13622afab5e9f8603afe42356.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b36a1938df65db2b68bf7595b97bafa2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/520b756756c7e6236056c3f490dafa5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/665ffcdb7c57534dc184cc840471f2f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e964a2e18a9ccba84979e7517f28c672.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/269b7463b79b93b9a80a4ebe651771c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d9b0d3e2d4b2add12c716e35bdce15b.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/b8ddee1d75e781cdb29811f008400c0b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8ddee1d75e781cdb29811f008400c0b.png)
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名校
【推荐1】如图,在平面直角坐标系
中,抛物线
过点
,
和
,连接
,
为抛物线上一动点,过点P作
轴交直线
于点M,交x轴于点N.
的解析式;
(2)如图1,连接
,当
是直角三角形时,求m的值;
(3)如图2,连接
,当
为等腰三角形时,求m的值;
(4)点P在第一象限内运动过程中,若在y轴上存在点Q,使得以O,P,Q为顶点的三角形与以B,C,N为顶点的三角形相似(其中点P与点C相对应),请直接写出m的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee31829d0d4d5f779a957d7df8058ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a90385c676848de67293e3ed6bc000fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45d57173ef4cd72eb270686875dfd623.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f605a7fc2f1064fe14882ee426839db.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7720309624a5e413f5e612bd492effe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/91d2d92a01447bbb2a6d0e4397bb9049.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0522607eeb42a499c0f5b040bbdd42e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
(2)如图1,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f258e16c4c882a60995fb9926626d554.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86a360a2f662b9996c2c7980b8672fcc.png)
(3)如图2,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf3369e0ea90e8d5cf4b6b3c45c0fd8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/699ccb1d8c438f5bc838cd3c55dacda0.png)
(4)点P在第一象限内运动过程中,若在y轴上存在点Q,使得以O,P,Q为顶点的三角形与以B,C,N为顶点的三角形相似(其中点P与点C相对应),请直接写出m的值.
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解题方法
【推荐2】如图,抛物线y=x2+bx+c交x轴于A、B两点(点A在点B的左侧),交y轴于点C(0,5),连接BC,其中OC=5OA.
(1)求抛物线的解析式;
(2)如图1,将直线BC沿y轴向上平移6个单位长度后与抛物线交于D、E两点,交y轴于点G,若点P是抛物线上位于直线BC下方(不与A、B重合)的一个动点,过点P作PM∥y轴交DE于点M,交BC于点H,过点M作MN⊥BC于点N,求PM+NH的最大值及此时点P的坐标;
(3)如图2,当点P满足(2)问条件时,将△CBP绕点C逆时针旋转α(0°<α<90°)得到△CB'P',此时点B′恰好落到直线ED上,已知点F是抛物线上的动点,在直线ED上是否存在一点Q,使得以点C、B′、F、Q为顶点的四边形为平行四边形?若存在,直接写出点Q的坐标,若不存在,请说明理由.
(1)求抛物线的解析式;
(2)如图1,将直线BC沿y轴向上平移6个单位长度后与抛物线交于D、E两点,交y轴于点G,若点P是抛物线上位于直线BC下方(不与A、B重合)的一个动点,过点P作PM∥y轴交DE于点M,交BC于点H,过点M作MN⊥BC于点N,求PM+NH的最大值及此时点P的坐标;
(3)如图2,当点P满足(2)问条件时,将△CBP绕点C逆时针旋转α(0°<α<90°)得到△CB'P',此时点B′恰好落到直线ED上,已知点F是抛物线上的动点,在直线ED上是否存在一点Q,使得以点C、B′、F、Q为顶点的四边形为平行四边形?若存在,直接写出点Q的坐标,若不存在,请说明理由.
![](https://img.xkw.com/dksih/QBM/2021/4/10/2696580968013824/2696993481777152/STEM/ca0087ee88e14bcfb5eb52db76b2224f.png?resizew=463)
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