1 . 综合与实践
在学习了“特殊平行四边形”后,兴趣小组的同学发现了这样一个问题:如图1,已知正方形
,
为对角线
上一动点,过点
作垂直于
的射线
,点
在射线
上,且
,连接
.通过观察图形,直接写出
与
的数量关系: .
(2)【类比探究】
兴趣小组的同学在探究了正方形中的结论后,将正方形换成矩形继续探究.如图2,已知矩形
,
,
,
为对角线
上一动点,过点
作垂直于
的射线
,点
在射线
上,且
,连接
.请判断线段
与
的数量关系,并说明理由.
(3)【拓展应用】
在(2)的条件下,点
在对角线
上运动,当四边形
为轴对称图形时,请直接写出线段
的长: .
在学习了“特殊平行四边形”后,兴趣小组的同学发现了这样一个问题:如图1,已知正方形
![](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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f6218fe7c62cd2dd2b7c70b8f91a61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(2)【类比探究】
兴趣小组的同学在探究了正方形中的结论后,将正方形换成矩形继续探究.如图2,已知矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4630b9ca3d20635161fa8fbf812b9128.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f26ad70d2b3aac8604834d57dfc59bb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56f6218fe7c62cd2dd2b7c70b8f91a61.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
(3)【拓展应用】
在(2)的条件下,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49404e6b19a342401a80d9feb1144605.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
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2 . 【问题呈现】
和
都是直角三角形,
,
,
,连接
、
,探究
、
的位置关系.
【问题探究】
(1)如图①,当
时,判断
、
的位置关系,并说明理由.
(2)如图②,当
时,
、
的位置关系为______.
【拓展应用】
(3)当
,
,
时,将
绕点C旋转、使A,D,E三点恰好在同一直线上,直接写出
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41663d9e5e18891475aeaa98794f33d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0fc75953bf5dcfa4af308c34bf9952d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f99100a3ec33eecf804076f76c905359.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4818e07559cd9ec3810dcb4d0cdb6a92.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
【问题探究】
(1)如图①,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf0086b054ef120408acac806a1b1318.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
(2)如图②,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e94f16d5ed858699bfea5039a7bf8ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
【拓展应用】
(3)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4590b1db8451f30e1021e68b8359c1e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a19338598965bb3856cdd0236bbf694.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a88770433e9897d78d28ccfaab8b506.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d631f45bc652539853f236952afa5bbf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/6/617a843d-cc61-479e-b34a-9ea7cac75f7f.png?resizew=791)
您最近一年使用:0次
3 . 问题提出
如图1,点
是等腰
底边
上一点, 点
,
在
同侧,且
,
,取
的中点
, 连接并延长
,
交于点
,探究
与
的数量关系.
(1) 先将问题特殊化, 如图
,当
时,直接写出
与
的数量关系;
(2)再探究一般情形.如图1,证明(1)中的结论是否仍然成立.
问题拓展:
(3)如图
,
不是等腰三角形, 点
是
边上的点, 连接
,
, 已知
于
,
, 且
,直接写出
的值.
如图1,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3182db896bc2462331796e2a6108363.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d7197b9c49f0585f548621a771d5b7bb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/57234018370c5f07523fb3c918eebf20.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d0213c5787a5a6b38d11bceca5567f67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/971b23e3ee827018557d6c88edd5369a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d87448e09eaa816e50ae92d111d5ded6.png)
(1) 先将问题特殊化, 如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ed5034a973110e2a6eb2e7d5699c24f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d87448e09eaa816e50ae92d111d5ded6.png)
(2)再探究一般情形.如图1,证明(1)中的结论是否仍然成立.
问题拓展:
(3)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/365822bd3945e6a3e871ca979c84cc12.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4d260c4df7b0dc180af6980d21f3371.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a3d223fd53549f2533a3cf914d399c8d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97bac88bb6ad658f5a896c3204086b1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/25990a4258e7d2e45aeec37622063b35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61f9215613b7b05217ed77268c238391.png)
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4 . 综合与实践
问题背景:
小明在研究直角三角板时,发现含有
的直角三角板具有一些特殊性质,于是他做出了如下探究.
初步发现:
如图①,在
中,
,
,作
的垂直平分线
交
于点D,交
于点E,连接
,则
______
,线段
,
,
的数量关系是________________;
事实上,在含
的直角三角形中,两个锐角恰好是2倍关系,而对于任意一个三角形,若有两个角存在2倍关系,这样的三角形还能具有上面发现的结论吗?带着这个疑问,小明又研究了下面一个三角形,请结合小明的发现,一起探究一下吧.
如图②,在
中,
,
,
,作
交
于点D.
;
(2)求
的面积;
拓展应用:
像这样的三角形,我们不妨称之为“倍半角三角形”.如图③,在菱形
中,
,
,
是对角线,点E,F分别在
,
上,且
.
,
,用含x的代数式表示y;
问题背景:
小明在研究直角三角板时,发现含有
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
初步发现:
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/56d3c9ce32b721995f355eea411340e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db90c0feb35aa54464f1303be9118f37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83873a9d782f2588c5eedbfe73f9bc2f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9e52a8f07834cbbbe4224962672fbbb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
事实上,在含
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
如图②,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f83da4fc72ea219e514566d4c190b6d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7099026716ee1821dd7d9f157dc055f5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9060f03b9ee41d70d135b1e1a8902ce9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e3361ca593cd2159bf8f8545a30d3dde.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
拓展应用:
像这样的三角形,我们不妨称之为“倍半角三角形”.如图③,在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b0ce914a50f5376766d5065e5f13d96.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6618b4d41917e695da2356c016242ba4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e2c30f73c718bde8352055a14987fc15.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b290efb9f94c6d30cffb5973a78d9ec.png)
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5 . 【发现与思考】
如图①,在矩形
中,对角线
与
相交于点
,点
是
中点,连接
,
,
与
交于点
,
,
.
(1)直接写出线段
、
的长度:
,
;
(2)直接写出线段
与
的比值:
;
【方法与探究】
如果将【发现与思考】中的“在矩形
中”这一条件变得更为一般化,改为“在平行四边形
中”——如图②,那么条件变了,线段
与
的比值是否保持不变?请说明理由;
【拓展与应用】
如图③,在
中,中线
与中线
相交于点
,点
是
的中点,连接
并延长交
于点
,若
,
,则请直接写出线段
的长度:
.
如图①,在矩形
![](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/1dde8112e8eb968fd042418dd632759e.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/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07160f14b3b453bebb64cb2bf96dc85a.png)
(1)直接写出线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a299d2b999568e80be8005565ba209a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1a903538428f3dea2713fe721ce4f4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32217457c4e96e2ef155cf15c1b65d97.png)
(2)直接写出线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/413f7649b9e569d314f44ea9839548bf.png)
【方法与探究】
如果将【发现与思考】中的“在矩形
![](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/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
【拓展与应用】
如图③,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53b95463a97c60db3250cb641bf6523d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a8a7b5adfcac0f46a4cd19da4ebb4a2b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26c69a9feb14d4b9ccba6ae42837fd73.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/3/50cd197d-8623-4151-bd2a-f9a5824c50d2.png?resizew=369)
您最近一年使用:0次
6 . (1)问题发现:如图1,在
和
中,
,
,连接
,填空:
;
;
(2)类比探究:如图2,在
和
中,
,连接
交
的延长线于点M,请判断
,并说明理由;
(3)拓展延伸:如图3,在(2)的条件下,将
绕点O旋转至点C与点M重合,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd4ae5d33fdc0a501b876dcfc6b5b54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
,填空:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4807ca16360c0cca436e59d4be98f626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2255c5ec3e056f67109c29e56cf8aa74.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1397f20656d3a4117f3145102105244.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b521c99e1f87bfb16c2060393482d71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5b0594893c40c5578051097ccad61e0.png)
(2)类比探究:如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3fe95f656b98b53f71a9d72bf0c9a4b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4807ca16360c0cca436e59d4be98f626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a20118349af7f5d14f8ef63e3a695dd.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/11c0fed1bec279a67db01c918711d4f5.png)
(3)拓展延伸:如图3,在(2)的条件下,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4807ca16360c0cca436e59d4be98f626.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd4ae5d33fdc0a501b876dcfc6b5b54.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c154be32bfa30e4c463c6085b0c8912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ae248960d8c1677cf948f8251275e863.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/18/f1810f13-84c8-4b42-b007-e3f1c457d825.png?resizew=550)
您最近一年使用:0次
7 . 综合与实践:
【课堂探究】在一堂主题为“几何翻折问题的拓展与深化”的数学公开课上,王老师给同学们每个人下发了一张平行四边形纸片,并且给出了一些数据(如图1),并提出,其中平行四边形两边所成的锐角的正切值为
.
(1)为了课堂的顺利进行,王老师先让同学们求出了
_____°;
【实践操作】数学课开始后,王老师让同学们在小组内先动手实践操作,对平行四边形进行翻折探究翻折后图形的性质.在同学们动手实践的过程中,王老师随机挑选了一个小组的操作成果(如图2)并进行展示,接着将其抽象为了一个数学问题.
(2)如图2,在(1)的条件下,点
在
上运动,点
在
上运动,点
为
沿着
翻折后点
的对应点.求证:当M是
中点时,点
落在
边的中线上;
【思考讨论】同学们经过思考后,纷纷给出了自己对于这个问题的看法.王老师非常高兴,继续将问题进行了深入.
(3)如图3,连接
,
,
,延长
至
,连接
,
.若
,
,求证:
平分
;
【学以致用】同学们经过本课的学习,对图形再次进行深入挖掘,在原来的基础上设问:
(4)在(3)的条件下,请直接写出
的正切值.
【课堂探究】在一堂主题为“几何翻折问题的拓展与深化”的数学公开课上,王老师给同学们每个人下发了一张平行四边形纸片,并且给出了一些数据(如图1),并提出,其中平行四边形两边所成的锐角的正切值为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d599cb4a589f90b0205f24c2e1fa021e.png)
(1)为了课堂的顺利进行,王老师先让同学们求出了
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/386cda9f96731510d52ec18199e18286.png)
【实践操作】数学课开始后,王老师让同学们在小组内先动手实践操作,对平行四边形进行翻折探究翻折后图形的性质.在同学们动手实践的过程中,王老师随机挑选了一个小组的操作成果(如图2)并进行展示,接着将其抽象为了一个数学问题.
(2)如图2,在(1)的条件下,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.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/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/033e44163b3b8e0813685744ee4da07a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
【思考讨论】同学们经过思考后,纷纷给出了自己对于这个问题的看法.王老师非常高兴,继续将问题进行了深入.
(3)如图3,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ffb952f86442845da723fd291564484.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab41cce6eb2d3058a644314865d16548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/943712a5e96b16cc15d775cc4687237e.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/c4b564c923bdec474b6779a1f8482f99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c63e36329f5e0979f5ee776ac5d06327.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c42c394431e2c46db0936a50efb780a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73e52abc8bcfbc43c80b221f60a3e45b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/356d4349bc0500c3b20b57012a440658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3ed24d7acba0b3a7ebf4d520296424a3.png)
【学以致用】同学们经过本课的学习,对图形再次进行深入挖掘,在原来的基础上设问:
(4)在(3)的条件下,请直接写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f7dca1b3ff383165df0de276308e833.png)
您最近一年使用:0次
8 . 【提出问题】
如图1,在
中,
,
,求
的最小值.
下面是小明、小红两位同学关于本题不同角度下的部分思维过程:小明:从代数角度看,设
,表示出
或者
,利用函数知识![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
小红:从几何角度看,延长
到点
,使得
,则
,连接![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2245611874cef35d5769a5f5968eef4.png)
【解决问题】
求AC的最小值.(可参考小明与小红的思路)
【深入探究】
如图2,
,
,点
从点
出发沿线段
向点
匀速运动,同时点
从点
出发沿射线
匀速运动,点
的速度是点
的两倍,连接
,取
的中点
,连接
,在
、
运动过程中,线段
的最小值是 .
【拓展提升】
如图3,
,
,点
从点
出发沿线段
向点
运动,同时点
从点
出发沿射线
匀速运动,点
的速度是点
的两倍,当点
到达点
时,点
停止运动,连接
,点
是线段
上一点,且
,连接
,在
、
运动过程中,求线段
的最小值.
如图1,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60d9142db4dd2ef151bf3d4a63afb61e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c36e46a71ff0988a74528243f1b77337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
下面是小明、小红两位同学关于本题不同角度下的部分思维过程:小明:从代数角度看,设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8c7a3d679b4dae63575903387a76ce45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/648cf8ab7773bbd7320ce7c51188ff2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
小红:从几何角度看,延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4dd6e7c88601b4c50b06f1238fdd95d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e673ef2d48215ca84a48377f17d6df00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2245611874cef35d5769a5f5968eef4.png)
【解决问题】
求AC的最小值.(可参考小明与小红的思路)
【深入探究】
如图2,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fcd0ced286a0fbc7e4862f8147264277.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.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/d40b319212a7e7528b053e1c7097e966.png)
【拓展提升】
如图3,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e918b70b02a73685e3c536c7f380e2c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efc6e4b936d7a800e839a30c3839574d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0b68df477b3ee45ac0f725db00d465a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7561e04b4e4cf0ce28f81e4c198c1f39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7bf58e5fea1189b33cf55d86335452f.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/a7bf58e5fea1189b33cf55d86335452f.png)
您最近一年使用:0次
9 . (1)问题发现
如图1,在等边三角形
中,
是
边上的中线,点D是线段
上一动点(不与点A重合),以
为边作等边三角形
,连接
.填空:
①
的值为________;
②
的度数为________.
(2)类比探究
如图2,在等腰直角三角形
中,
,
,
是
边上的中线,点D是线段
上一动点(不与点A重合),以
为边作等腰直角三角形
,
,连接
.请判断
的值与
的度数,并说明理由;
(3)拓展延伸
如图3,在正方形
和正方形
中,
,点E在对角线
上,请直接写出
面积的最大值及此时线段
的长度.
如图1,在等边三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79ff575e55857af133edb24c8e61504f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
①
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274ae3f70122c99addb6df12405bac91.png)
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d29e7c4d6216690152430b09f9cf0c2.png)
(2)类比探究
如图2,在等腰直角三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7bef5239ddbb0972700ce01daf9ee7cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9c06154cae3bf7a8ce5a1e97a7380875.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10fc7991ea17d54ff5f4445ac5699463.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d831c9ba6d18774540e2aaa29132fec9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274ae3f70122c99addb6df12405bac91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d29e7c4d6216690152430b09f9cf0c2.png)
(3)拓展延伸
如图3,在正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b1f13007c6d134c50004c62dc240707.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ced06b71073e1bb777f326f06016ce17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19697acb629aed98c7733bcffd56fd0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
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10 . 综合与实践
问题情境:在矩形
中,E为边
上一点,且
,过点D作
于点P,交
于点M,将
沿
折叠,点A的对应点N恰好落在
边上.
如图1,若
,
与
的数量关系是 ,
的值是 ;
(2)类比探究
若
,写出
与
的数量关系及
的值,并就图2的情形说明理由;
(3)拓展应用
在(2)的条件下,将
绕点D在平面内自由旋转,得到
,当
的一条直角边与
平行时,连接
,请直接写出
的面积.
问题情境:在矩形
![](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/08f57b8380e1c9d059087e2bd121153f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b67e608f3ab645c2a9b88cdf7a142791.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3da7bcea5a45eeae211f5851f12a7517.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fd17a66a2af938c89e46f22e4d893b1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3570a95f68349fcd9417fcda62e78e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8b77b30405133bc2e517cf0db64df65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8783e6e04b752b55724a0688e343a89f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d0065d620aaf8742c7928574b81913a.png)
(2)类比探究
若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff9b24557f60182cb2dcd66852e32f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8b77b30405133bc2e517cf0db64df65.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8783e6e04b752b55724a0688e343a89f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d0065d620aaf8742c7928574b81913a.png)
(3)拓展应用
在(2)的条件下,将
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dff8f3c6432e7a98f14e94b5cc57fb0a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e6664b42413839b9d1e596ab19cc597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e6664b42413839b9d1e596ab19cc597.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a630858806f18ca3aafd1ff1383531f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33596fbe30fb3868f5f7af158f57967b.png)
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