1 . 李老师善于通过合适的主题整合教学内容,帮助同学们用整体的、联系的、发展的眼光看问题,形成科学的思维习惯.下面是李老师在“图形的变化”主题下设计的问题,请你解答.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/17/d91ff9b5-84cc-40b3-9f37-fb75e1ac4da6.png?resizew=517)
(1)问题背景
如图
,正方形
中,点
为
边上一点,连接
,过点
作
交
边于点
,将
沿直线
折叠后,点
落在点
处,当
时,
____________;
如图
,连接
,当点
恰好落在
上时,其他条件不变,则
____________;
(2)探究迁移
如图
,在(
)的条件下,若把正方形
改成矩形
,且
,其他条件不变,请写出
与
之间的数量关系式(用含
的式子表示),并说明理由;
(3)拓展应用
如图
,在(
)的条件下,若把正方形
改成菱形
,且
,
,其他条件不变,当
时,请直接写出
的长.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/2/17/d91ff9b5-84cc-40b3-9f37-fb75e1ac4da6.png?resizew=517)
(1)问题背景
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b5f0805cc7a96c866df10b667ac4851e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47b13fff157b3ff2b7d80c0cfb3e7a21.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f331b8b4f9fc690b2e4785c8dd0527dc.png)
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7c314398e26ffc7164b82946eeb4273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9af9650e51fef805788fe4fea3cfb527.png)
(2)探究迁移
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](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/a52a55bd36f9d239416d7c8fa5d78bc3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71fe246270d1277f9eb2bf15af22e83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(3)拓展应用
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](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/284e282bb1d9fbf8634b3506ee5358ab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27f39dacbcf8ca8364b47ac277beec70.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c628297022d4dee189225c9441877539.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71fe246270d1277f9eb2bf15af22e83.png)
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2 . 【问题呈现】
如图,
和
是有公共顶点的直角三角形,
,点P为射线
、
的交点.探究
,
的位置关系.
(1)如图1,若
和
是等腰直角三角形,求证:
;
(2)如图2,若
,(1)中结论
是否仍然成立?请说明理由;
【拓展应用】
(3)在(1)的条件下,
,
,将
绕点A旋转,使点E恰好落在线段
上,请直接写出此时
的长度.
如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26813466e2ee49a493881a4384fc8748.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/63a253c7fdf589ee3dece13d5b5b5732.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(1)如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/945efbd2a73b8f37b115b56f4f3ea937.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9da18b1095150726af9c9cef6015b938.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/945efbd2a73b8f37b115b56f4f3ea937.png)
【拓展应用】
(3)在(1)的条件下,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/267ace52b64e1e7dfc5211e033255b7d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
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3 . 在
中,
,点
在直线
上,直线
与
的夹角为
, 且
,分别过点
,
作直线
的垂线,垂足分别为
,
.
如图
,若
,则
的度数为________,
的值为______;
(2)【问题探究】
如图
,若
,判断
的值是否发生变化?并说明理由;
(3)【拓展延伸】
如图
,
,
交于点
, 点
在线段
上 ,
,
,求线段
的长.
![](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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76bc8e50eeccced1d7f3bc52e09a7d03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7be10d1176bcc1ad83ebb7758db1236a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1c9ae241fd78126274c65e17990c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45723dce96b74f027adcb1a4ecd19db8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1682a8242948cc1d76eddef3e50836f.png)
(2)【问题探究】
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c650fe55b7603f106c53ca2423451c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26f46bb7d89e2365a06c4da9c2e46d0d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1682a8242948cc1d76eddef3e50836f.png)
(3)【拓展延伸】
如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f82e415812cca9545611c0faa0c01b1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b9bfe825bce9cb9b8a79a9a9a5755c4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/833cfda415649b832cc136caed392753.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
您最近一年使用:0次
4 . 某数学兴趣小组在数学课外活动中,对多边形内两条互相垂直的线段做了如下探究:
(
)如图
,在正方形
中,点
分别是
上的两点,连接
,
,则
的值为__________.
(
)如图
,在矩形
中,
,
,点
是
上的一点,连接
,且
,则
的值为__________;
【类比探究】
(
)如图
,在四边形
中,
,点
为
上一点,连接
,过点
作
的垂线交
的延长线于点
,交
的延长线于点
,求证:
.
【拓展延伸】
(
)如图
,在
中,
,
,
,将
沿
翻折,点
落在点
处得
,点
分别在边
上,连接
,
,
.求
的值.
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a373959bb9026f8a09845c0b828bf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/137717f0e360e82319ced7e3a6c69b2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21cc3c5593dee883a54618c948ca8807.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9a128183b4d191a765ef19ff41aecc43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ada5e380d6b5b2066e34497e5f0baf0.png)
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78c1e75eb744576c32eed3f9d20f9558.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e673ef2d48215ca84a48377f17d6df00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c0fe1363ef62726e068550cf6b145596.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a45b04cc3e5adaeff6f9e01e29032803.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8775dcf60ad143bc59a58b000f374c56.png)
【类比探究】
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/41aba1711910c6f533cc94319104f4fd.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/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/21037e170bdbb322558e79c40c00b454.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.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/ccb545c1a90c841d0d25070cf62b64e3.png)
【拓展延伸】
(
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16834582e86f9c1030a4a4b316fab909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ce0d7095ddd69d6ceaf1065b1bc2c79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a306417e43670260e4b68a928a22071f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/53289e3083b05d556dd30197abc547ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73636989e83905f8800a865c2b608c43.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a373959bb9026f8a09845c0b828bf82.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/137717f0e360e82319ced7e3a6c69b2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1f87b02b744534ae1ed700d21fcceb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ada5e380d6b5b2066e34497e5f0baf0.png)
您最近一年使用:0次
2024-05-12更新
|
324次组卷
|
3卷引用:2024年广东省阳江市阳春市中考一模数学试题
2024年广东省阳江市阳春市中考一模数学试题2023学年贵州省铜仁市碧江区铜仁学院附属中学九年级下学期第5次模拟预测题(已下线)考前特训03 几何解答题探究综合压轴题-2024年中考数学二轮热点题型归纳与变式演练(全国通用)
5 . 综合与实践
问题情境:数学活动课上,老师将两个具有公共顶点的全等三角形按图1所示摆放,
,
,
,
.老师让各小组在此基础上展开探究.
延长,分别交
于点O和点F,试判断
与
的数量关系并说明理由;
深入探究:(2)善思小组固定
,将
绕点B逆时针旋转,如图2,当
时,
与
相交于点P,过点P作
于点Q,试判断
与
的数量关系并说明理由;
拓展延伸:(3)创新小组将图1中的
绕点B逆时针旋转,如图3,当
时,
与
相交于点M,过点M作
于点N.若
,
,请直接写出
的长.
问题情境:数学活动课上,老师将两个具有公共顶点的全等三角形按图1所示摆放,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/74db1f028c135803c5f56852ec330d01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/546f5e3bf6a3b60f85ea730a6cd7b36e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/226c231c1150e27c34eb787a46b02ea8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8915e8e775538d41debf1933102c6b86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17487118ab7a90fc8c91bf0870f3289e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e054d52ef2c878ea3097ef237f1d3801.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
深入探究:(2)善思小组固定
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abe4bdc5d9e833b23a1b916c06fc1a35.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/92b154270249b0ef54ddb26137b2681a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7944350f95810a29ca8c032bc1d6659e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
拓展延伸:(3)创新小组将图1中的
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abe4bdc5d9e833b23a1b916c06fc1a35.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e78ee2cb6caf7d7ee877796d18e53907.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/66fae8e33cd86fa8dab72704eaafe1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cd0763b031b7e6b6d87ce3554ac482d5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65fc998d33cdf37c272f79cfd64b7b99.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
您最近一年使用:0次
6 . 综合与探究.
【特例感知】
(1)如图(a),
是正方形
外一点,将线段
绕点
顺时针旋转
得到
,连接
,
.求证:
;
【类比迁移】
(2)如图(b),在菱形
中,
,
,
是
的中点,将线段
,
分别绕点
顺时针旋转
得到
,
,
交
于点
,连接
,
,求四边形
的面积;
【拓展提升】
(3)如图(c),在平行四边形
中,
,
,
为锐角且满足
.
是射线
上一动点,点
,
同时绕点
顺时针旋转
得到点
,
,当
为直角三角形时,直接写出
的长.
【特例感知】
(1)如图(a),
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6aa2b5e09f8ec785c59900a529390a02.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86d1125b47dec1c5e93143ee59ad862a.png)
【类比迁移】
(2)如图(b),在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d2c15801fee2405573677484f5dcfa4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32436704a722d5e568ff5c175bf3c662.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/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc5adb5eb60ae4435a12d93854066298.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4fb26d84907c923278ac4626a9d58947.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/4eedae8d316c76e3d0b451256de03fb9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/920ae90193d50ae60beafae6f2b4ebcd.png)
【拓展提升】
(3)如图(c),在平行四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/78b65065ec3a0cb4b050989165c003d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f26ad70d2b3aac8604834d57dfc59bb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/febc9a89d0d1c97b88c0f4acd32b4e67.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/905f306dc08525842a96cfcab32c1ce8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c02b54dc6b3e1bb6544f47d4c8743fcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39e210795a966bd5dc1293479ab1988b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4171e7f713d6b265d56b2662b7af57b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ff4e82ffd7c5c97839c48959b453b91e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
您最近一年使用:0次
名校
7 . 问题提出
(1)如图1,在正方形
中,E、F分别是边
和对角线
上的点,
.求证:
;
问题探究
(2)如图2,在矩形
中,
,E,F分别是边
和对角线
的点,
,
,求
的长;
拓展延伸
(3)如图3,在菱形
中,
交
的延长线于点G.E,F分别是线段
和
上的点,
,
,求
的长.
(1)如图1,在正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.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/345c91c9e67e6ee00a872078ab17f094.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f2a0ee47f2bfbf68c8a1c4da23894e3.png)
问题探究
(2)如图2,在矩形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32b4da8d254b7d76b7615d7f7cd824a3.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/6234f097c4a50340d86d0b9007d69eb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/051b3a459c945cc6ec4a17bc0e39566f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
拓展延伸
(3)如图3,在菱形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dfe8c741f56e946c1b454732f8faaa9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/67d822262ff00915910e5b87d81ad1ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4083c581c6027c4b2ae7e3b3749f485.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6234f097c4a50340d86d0b9007d69eb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c41f3062c3f910e418d0ac2bd62303de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://img.xkw.com/dksih/QBM/2024/1/29/3421543211474944/3427181465468928/STEM/9c4072c539ce481fa6dab995d30eb73a.png?resizew=481)
您最近一年使用:0次
2024-02-06更新
|
112次组卷
|
2卷引用:陕西省西安市未央区2023-2024学年九年级上学期期末数学试题
名校
8 . 综合与实践.
【问题发现】(1)如图1,在正方形
中,E为对角线
上的动点,过点B作
的垂线,过点C作
的垂线,两条垂线交于点F,连接
,求证:
.
中,E为对角线
上的动点,过点B作
的垂线,过点C作
的垂线,两条垂线交于点F,且
,连接
,求
的值.
【拓展延伸】(3)如图3,在(2)的条件下,将E改为直线
上的动点,其余条件不变,取线段
的中点M,连接
,
.若
,则当
是直角三角形时,请直接写出线段
的长.
【问题发现】(1)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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d96a5d40d0aea9f4398ca4d0fe9b0dd.png)
![](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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26fdd8e57562ba94e10e7f1d770826d0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4a12235fc709caecafb64248d2fb8001.png)
【拓展延伸】(3)如图3,在(2)的条件下,将E改为直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e69d2b798744645af88a4fa411344a83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f7bae5203f4b4acf23779114b3466e17.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8045b729b7f257861057191a8a2388.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
您最近一年使用:0次
9 . 综合与实践:
中,
是边
上一点,
于点
,
,
,
,求证:四边形
为正方形;
【实践探究】(2)小宇受此问题启发,逆向思考并提出新的问题:如图2,在正方形
中,
是边
上一点,
于点
,
于点
,
交
于点
,请探究线段
,
,
之间的数量关系并说明理由;
【拓展迁移】(3)小阳深入研究小宇提出的这个问题,发现并提出新的探究点:如图3,在正方形
中,
是边
上一点,
于点
,点
在
上,且
,连接
,
,请探究线段
与
的数量关系并说明理由.
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1893d78af450a5fa09810537adc2dfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee38668bd4d62fe28999499c5a978b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ecbb05825a4055d125626e81a1ac2e1c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dc6fa61fc614eeb32b387a35688948e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
【实践探究】(2)小宇受此问题启发,逆向思考并提出新的问题:如图2,在正方形
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1893d78af450a5fa09810537adc2dfa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ab4bf86fe0ed4bbc4c9e00c5d838ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ee38668bd4d62fe28999499c5a978b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/895dc3dc3a6606ff487a4c4863e18509.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c83f1f880e5ffbff036953acaca90c41.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/826c728050e3378921442ace20269ef6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
【拓展迁移】(3)小阳深入研究小宇提出的这个问题,发现并提出新的探究点:如图3,在正方形
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54ab4bf86fe0ed4bbc4c9e00c5d838ba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73465a1f9aa03481295bf6bd3c6903ac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6655e2fa64a32cd12fe0279afd65d73.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cbaff1c99c2773858fa998537137d608.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcdae78f4d3b8d8213ac3ac9a9567eb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
您最近一年使用:0次
名校
10 . 如图1,四边形
是平行四边形,点
是对角线
上一点,点
是
外一点,连接
、
和
,且
,
,
.
是菱形;
(2)【拓展延伸】如图2,连接
交
于点
,延长
交
于点
,求证:
;
(3)【实践应用】我校生态社在校内的蔬菜种植活动硕果累累,备受关注.如图3所示的一块正方形
种植区被
、
、
分割种植着不同植物,经测量得
,
.现学校决定延长
交
于点
,以
为边长,在该种植区的左边再开辟一块小正方形新区域种植更多蔬菜,求新区域的面积.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc56c77464a17a1e97b568762a3e2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac679f877d3e3b3acef1f1f8e3654b4b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad4c5d9a889842bb78bcf12f0909c4c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b4bbe928cbfa7ccd2d93a08f7a92861.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(2)【拓展延伸】如图2,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a74fa1d77910529e4eda3b2d70f40a1.png)
(3)【实践应用】我校生态社在校内的蔬菜种植活动硕果累累,备受关注.如图3所示的一块正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5634a208294350f8ec151cd19b6d3278.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f8e5f736b1195fef1d2d300168a795f1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a3df7035b26938612586bbd963914fd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34d795825a51d2a1954018548255f91c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4505508b3e36db64a207dcdaf8eb22dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84d454c82d9e52747563d47b68099249.png)
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