从反思中总结基本活动经验是一个重要的学习方法.例如,我们在全等学习中所总结的“一线三等角、K型全等”这一基本图形,可以使得我们在观察新问题的时候很迅速地联想,从而借助已有经验,迅速解决问题.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/1/e08d1ac3-f6df-4a5a-ad44-8b1fb96a8523.png?resizew=526)
(1)如图1,在平面直角坐标系中,四边形
是正方形,且
,点E是线段
延长线上一点,M是线段
上一动点(不包括点O、B),作
,垂足为M,且
.设
,请你利用基本活动经验直接写出点N的坐标 (用含a的代数式表示);
(2)基本经验有利有弊,当基本经验有利于新问题解决的时候,这是基本经验的正迁移;当基本经验所形成的思维定势局限了新问题的思考,让新问题解决不出来的时候,这是基本经验的负迁移.例如,如果(1)的条件去掉“且
”,加上“交
的平分线于点N”,如图2,求证:
.如何突破这种定势,获得问题的解决,请你写出你的证明过程.
(3)如图3,请你继续探索:连接
交
于点F,连接
,下列两个结论:①
的长度不变;②
平分
,请你指出正确的结论,并给出证明.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/12/1/e08d1ac3-f6df-4a5a-ad44-8b1fb96a8523.png?resizew=526)
(1)如图1,在平面直角坐标系中,四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ab158dd9e96abcae0f115f54fa66c5b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7b437423e59ede2bf276c33d0fd644f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2a9be2f8ba5170eaf0bc4604e021706.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5bba25a314fb799efba2895cb1f1f56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d93f2968ff78d56d7c71bc7bd33101a.png)
(2)基本经验有利有弊,当基本经验有利于新问题解决的时候,这是基本经验的正迁移;当基本经验所形成的思维定势局限了新问题的思考,让新问题解决不出来的时候,这是基本经验的负迁移.例如,如果(1)的条件去掉“且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5bba25a314fb799efba2895cb1f1f56.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2d29e7c4d6216690152430b09f9cf0c2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b920f015c63f066a8ae00e3066b50e20.png)
(3)如图3,请你继续探索:连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3533837e3d08c461dea031a44e5424d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2273ae1ee99cec9c1304323bc9ebf75f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2273ae1ee99cec9c1304323bc9ebf75f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7032cd1fe3619ae86ef81b1c4363b1d3.png)
2022八年级上·全国·专题练习 查看更多[2]
(已下线)专题13 一线三等角模型证全等-【微专题】2022-2023学年八年级数学上册常考点微专题提分精练(人教版)(已下线)专题20 平面直角坐标系中的正方形-【微专题】2022-2023学年八年级数学下册常考点微专题提分精练(人教版)
更新时间:2022-11-27 16:12:59
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相似题推荐
解答题-问答题
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【推荐1】如图,在直角坐标系中,已知
,
,
三点.
![](https://img.xkw.com/dksih/QBM/2021/6/29/2753531359092736/2753595720638464/STEM/8f662d61a68e46e6abf1c6f415672d1f.png?resizew=234)
(1)求四边形
的面积;
(2)是否存在点
,使
?若存在,求出点
的坐标.若不存在,请说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32366143230ca122894a4bada7c7b96d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08ef03f452410ab19c6246567c427178.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b7089a4e568ed92b1230af589f4d47be.png)
![](https://img.xkw.com/dksih/QBM/2021/6/29/2753531359092736/2753595720638464/STEM/8f662d61a68e46e6abf1c6f415672d1f.png?resizew=234)
(1)求四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be9040cff7e9e20dd2817ad9888237d8.png)
(2)是否存在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d16c131b998009c4fee395adfb02a9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaa9fef9de4ffb371e93f71eadd13aba.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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【推荐2】小明在学习《勾股定理》时,利用平面直角坐标系在研究两点的距离时,通过数形结合发现(如图),平面内的任意两点
,
的距离,满足.经小明查阅资料得知,以上发现是成立的.在平面直角坐标系中,
叫
,
两点的距离公式.请你根据数形结合的思想和所学知识,完成以下问题:
![](https://img.xkw.com/dksih/QBM/2023/5/21/3242295521984512/3259783928520704/STEM/e27fb1817a384378ab03dde5b0b9cdc2.png?resizew=292)
(1)直接写出过
、
两点的距离为__________.
(2)若点
在直线
上,点
到
和
的距离相等,请求出点
的坐标.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa214e77f478029fa3472d0caef749b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12a3efb79f35db8448f3391252ab7d4e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df332f01628130c084fd46aaca0a4b7.png)
![](https://img.xkw.com/dksih/QBM/2023/5/21/3242295521984512/3259783928520704/STEM/e27fb1817a384378ab03dde5b0b9cdc2.png?resizew=292)
(1)直接写出过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5eb44cd13c6f3c7adcd5dc7844b0826.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9074e9f074ee6a848474791af207ad9.png)
(2)若点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d77f5191798242b7b9b88a75e17e4425.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1cf8af6e62058cc4e2d83d5da7f4c68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8744706bd401eb5965a3141e6831bab3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
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【推荐1】数学模型学习与应用:
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/6/6e5c12b7-e5a5-47c7-a3f7-9019dbbf3064.png?resizew=551)
(1)【模型学习】:如图
,
,
,
于点
,
于点
由
,得
;又
,可以通过推理得到
≌
,进而得到
______,
______
我们把这个数学模型称为“一线三等角”模型.
(2)【模型应用】:如图
,
为等边三角形,
,
,求证:
;
(3)【模型变式】:如图
,在
中,
,
,
于点
,
于点
,
,
,则
______.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/8/6/6e5c12b7-e5a5-47c7-a3f7-9019dbbf3064.png?resizew=551)
(1)【模型学习】:如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2032d08bfae318bf1669d27aa8e6366f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/adb32dae6823f572ffa028d684733c37.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ecdd28c3de7d4bf81d2ceee16f76cf5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abc0613441dac089635db3d6efd4aeb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7186b756542b54106635d9256528aa91.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/94f32b4b554381071e828a0946d7fce4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8835e928bfe044009c508cc5c19bf611.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/da2c203d9f8d3af78921dc9d3de25bc2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d189dd249d31d4b0497def200debb02d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/634d6f7a95c2c1484be3324e5a53953c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/140e540849151c98a71e423aa443b39e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c90282d4a37c9a20620d4bbb0c263cae.png)
(2)【模型应用】:如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e74451cb021d33293b020876f90e433f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7867957591914f5c08a3897570dc29a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07601cea58f3c4a194798a70716da86d.png)
(3)【模型变式】: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/82da6815bc213dfd78c2f77cd7ded8ed.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c287bb15494297df234f700f9f3a90d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17e25aeda8821415ca7aeb86a72de6a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4f5f991a84ee63e01ad88c4620a38500.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/52a71070d7c38fa7c3b7581c5008095d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/295a9703ec54bba72347545cc966f337.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03dc287c7bc96dbb85d11795875582a7.png)
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【推荐2】如图,在
中,
是它的一条对角线,作
的垂直平分线
,分别交
于点
.
(1)求证:
;
(2)连接
,若
,求
的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5138a9f70d5e8b0580e30fef6eb7baef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12d7fcf5cac4fe80cf700c90bf423d7a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48c58d446299b1627baa8ed108eeb2d9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/8/6/eb3525f0-ad7d-438a-ba9e-564548809510.png?resizew=154)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d8ce1475f537b4ad21775bfaa16daa0c.png)
(2)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4b7c547efa568cc49326ae560a38744a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/05e8589b90b9f931af7dc863af83fb71.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c962fe4f47732b8e6e83d17ff2b9af13.png)
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名校
【推荐1】证明文字命题:对角线互相垂直的矩形是正方形.(画出图形、写出已知、求证与证明)
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【推荐2】如图,已知四边形ABCD中,AD=4,CD=3,AB=AC,
![](https://img.xkw.com/dksih/QBM/2015/7/9/1573866844438528/1573866850426880/STEM/f54518bfb25b49c79f5b9b1ad0ae1baa.png)
![](https://img.xkw.com/dksih/QBM/2015/7/9/1573866844438528/1573866850426880/STEM/48e623ca98fc4d379bcc204f8cbdc8a7.png)
(1)如图1,若∠CAB=60°,∠ADC=30°,求BD的长;
(2)如图2,若∠CAB=90°,∠ADC=45°,求BD的长.
![](https://img.xkw.com/dksih/QBM/2015/7/9/1573866844438528/1573866850426880/STEM/f54518bfb25b49c79f5b9b1ad0ae1baa.png)
![](https://img.xkw.com/dksih/QBM/2015/7/9/1573866844438528/1573866850426880/STEM/48e623ca98fc4d379bcc204f8cbdc8a7.png)
(1)如图1,若∠CAB=60°,∠ADC=30°,求BD的长;
(2)如图2,若∠CAB=90°,∠ADC=45°,求BD的长.
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【推荐1】如图1,已知正方形
的边长为1,点
在边
上,若
,且
交正方形外角的平分线
于点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/13/2e0a4b62-3526-4367-9598-cc8b16a45ae4.png?resizew=397)
(1)如图1,若点
是边
的中点,
是边
的中点,连接
,求证:
.
(2)如图2,若点
在线段
上滑动(不与点
,
重合).
①在点
滑动过程中,
是否一定成立?请说明理由;
②在如图所示的直角坐标系中,当点
滑动到某处时,点
恰好落在直线
上,求此时点
的坐标.
![](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/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7521601ac1fbc4a52d4fe44be9c8e88c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4cae70b8a9d2d2e96dea62c00ced04b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/13/2e0a4b62-3526-4367-9598-cc8b16a45ae4.png?resizew=397)
(1)如图1,若点
![](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/ac047e91852b91af639feec23a9598b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b46c607b3deac746c0ef3389ad8f65c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3767efa8adfb71001ad39df9560cbf6a.png)
(2)如图2,若点
![](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/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
①在点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3767efa8adfb71001ad39df9560cbf6a.png)
②在如图所示的直角坐标系中,当点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c60ecd5f53b3469726c8cb09e86ff20b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
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【推荐2】如图1,在正方形ABCD中,点E是边AB上的一个动点(点E与点A,B不重合)连接CE,过点B作BF⊥CE于点G,交AD于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/25/ba00b2cf-8389-4b37-b40d-d27c529e7819.png?resizew=504)
(1)求证:△ABF≌△BCE;
(2)如图2,连接EF、CF,若CE=8,求四边形BEFC的面积;
(3)如图3,当点E运动到AB中点时,连接DG,求证:DC=DG.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/5/25/ba00b2cf-8389-4b37-b40d-d27c529e7819.png?resizew=504)
(1)求证:△ABF≌△BCE;
(2)如图2,连接EF、CF,若CE=8,求四边形BEFC的面积;
(3)如图3,当点E运动到AB中点时,连接DG,求证:DC=DG.
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