【阅读理解】截长补短法,是初中数学几何题中一种辅助线的添加方法.截长就是在长边上截取一条线段与某一短边相等,补短是通过在一条短边上延长一条线段与另一短边相等,从而解决问题.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/3/f050073b-8a08-497f-96be-24f5950395a6.png?resizew=417)
(1)如图1,
是等边三角形,点
是边
下方一点,
,探索线段
、
、
之间的数量关系.
解题思路:延长
到点
,使
,连接
,根据
,可证
,易证得
≌
,得出
是等边三角形,所以
,从而探寻线段
、
、
之间的数量关系.
根据上述解题思路,请写出
、
、
之间的数量关系是______,并写出证明过程;
【拓展延伸】
(2)如图2,在
中,
,
,若点
是边
下方一点,
,探索线段
、
、
之间的数量关系,并说明理由;
【知识应用】
(3)如图3,两块斜边长都为
的三角板,把斜边重叠摆放在一起,则两块三角板的直角顶点之间的距离
的平方为多少?
![](https://img.xkw.com/dksih/QBM/editorImg/2023/1/3/f050073b-8a08-497f-96be-24f5950395a6.png?resizew=417)
(1)如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bf44443e2078d660867b06f4948aeff.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96461af9193d5d7867ddf8fcbefcaec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c3f982d999937891eec4cb22d62e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdd9f345915ae742ed3dcd3f9678264.png)
解题思路:延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdd9f345915ae742ed3dcd3f9678264.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/efbc2b57a1d42e046628ce5466638a24.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b9f79d7939c88e9702962e5917cad290.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/34323265b942076ed69169b0be012e01.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8df43abe4a6c212a1a2a26b19a218409.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bb961bd7db3adb76af2d4cedb611bd7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a8b0d7e169313c2a2d5c278cf968b9a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aefe4a3e7a7fa195ed6a6712447639b7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d1ca3d89a7eb6e8bf9ce7c1b14b6024.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96461af9193d5d7867ddf8fcbefcaec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c3f982d999937891eec4cb22d62e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdd9f345915ae742ed3dcd3f9678264.png)
根据上述解题思路,请写出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96461af9193d5d7867ddf8fcbefcaec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c3f982d999937891eec4cb22d62e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdd9f345915ae742ed3dcd3f9678264.png)
【拓展延伸】
(2)如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7281b641656a5992abaafb4190ca9afc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1c8b77e0a8fdb0bbb32e1e25583ad62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6ab41054fa9ce51b68e78d9c0cf398d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/764509115979e9958101808383672ec0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b2baf410f949feec3ffbd032d1da5b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f96461af9193d5d7867ddf8fcbefcaec.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/93c3f982d999937891eec4cb22d62e5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8cdd9f345915ae742ed3dcd3f9678264.png)
【知识应用】
(3)如图3,两块斜边长都为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f2ccb10cec178445fc5060f7b651a9ef.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e430f13f42cf2d44aa0f0e20b959684f.png)
21-22八年级上·江苏扬州·阶段练习 查看更多[5]
江苏省扬州市江都区八校2021-2022学年八年级上学期12月月考数学试题(已下线)3.3 (附加2) 利用勾股定理解决折叠、展开等问题(练习)-2022-2023学年八年级数学上册同步精品课堂(苏科版)湖南省永州市宁远县冷水镇上宜中学2021-2022学年九年级上学期第三次月考数学试卷湖南省永州市宁远县上宜中学2021-2022学年上学期九年级第三次月考数学试卷山东省烟台市蓬莱区2022-2023学年七年级下学期期末数学试题
更新时间:2021-12-21 10:12:31
|
相似题推荐
解答题-证明题
|
适中
(0.65)
【推荐2】已知在
中,满足
,
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/16/73483baf-16ed-4769-9db3-cd96c84c457d.png?resizew=483)
(1)【问题解决】如图1,当
,
为
的角平分线时,在
上取一点
使得
,连接
,求证:
.
(2)【问题拓展】如图2,当
,
为
的角平分线时,在
上取一点
使得
,连接
,(1)中的结论还成立吗?若成立,请你证明:若不成立,请说明理由.
(3)【猜想证明】如图3,当
为
的外角平分线时,在
的延长线上取一点
使得
,连接
,线段
、
、
又有怎样的数量关系?请写出你的猜想,并对你的猜想给予证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b903796831a684296f24e836b1fa7770.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/9/16/73483baf-16ed-4769-9db3-cd96c84c457d.png?resizew=483)
(1)【问题解决】如图1,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934a9508c176f44bf58f88715bd98f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/90c251ed1472ba56f13a80abbfeb06c1.png)
(2)【问题拓展】如图2,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8c44fc4836a86d5c24c20e6807eec9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934a9508c176f44bf58f88715bd98f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6e490f703eb6c9bb1278c78ebc2d661.png)
(3)【猜想证明】如图3,当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/934a9508c176f44bf58f88715bd98f76.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/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
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【推荐1】阅读材料解决问题:在锐角
中,
的对边分别为
,作
于点
,在
中,
,
,在
中,
,
,即
,
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/9/c4d65db3-7e7b-4db2-8f02-e2ff9c81dd49.png?resizew=403)
(1)证明:
;
(2)如图二,求
(结果保留根号);
(3)如图三,在锐角
中,
,
,又
,垂足为
,
,求
的长度.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c9c1e84aaa7e1b5c1283075b36c72fb5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b5f215a42c4b7078d8d65923eb9980e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16834582e86f9c1030a4a4b316fab909.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7c7e2b1e5771ce36405586ca499d0a7c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51d92dba6cf728dd156c3647e70f8c95.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/10ce5d6ca24e249f7172e51c154509b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03477829f4acb56b7c2c1c0fd9816f68.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d91332994ffcc70181a0e226d8ecfcf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ca3ab19446cac7945490fd138b03bb80.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b14c36d9290b0a9d82be59196f0e62f8.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/9/c4d65db3-7e7b-4db2-8f02-e2ff9c81dd49.png?resizew=403)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6653d40c134c72d36f0bad667acb2eef.png)
(2)如图二,求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d07e786b38c88bc9d3cc458ba0222e79.png)
(3)如图三,在锐角
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bbdde27fe544ac1e77b6a6a0b2fde32.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f760ada0e0b34bdc9d60ba62d19f03b3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b757f0c42ae5c9a2d6a4b19e5877b27.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d71e6ea7333dbc78d0a7b9bc3892f940.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
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【推荐1】如图,点
是等边
内一点,将
绕点
逆时针旋转
得到
,连接
.
是等边三角形;
(2)若
,
时,求
的度数.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a7ffcd1925a2b1259221c6a476152f7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/260200d547998bcac50a4a491382e7f4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd6f6270ffb9ba9dcbfc795642e17ae2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/683c590673eece14fea3319c4fd5eb55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d4686f39b38d5b90309ee73ed89a0640.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d5963ad524682b8c3d6737b7e8bf4602.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9979b72539453dd5e64dbd2fe3e0e09c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d22df2977de56cc69be0c1e847653d7a.png)
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名校
【推荐2】如图1,已知
为等边三角形,点P、E分别是AB、AC边上一点,
,连接CP、BE交于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/21/f4e571f3-743e-4e3a-ab27-4a324c47fd48.png?resizew=389)
(1)求
的度数;
(2)如图2,将线段CP绕点C顺时针旋转120°得线段CQ,连接BQ交AC于点D,
①在图中找一个与
全等的三角形,并说明理由;
②探究BP、CD、BC的数里关系,并说明理由.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7447ddd43b6d3b5bc91775f8782ed50d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/21/f4e571f3-743e-4e3a-ab27-4a324c47fd48.png?resizew=389)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77282d5b97a60264cf15fb74b48e4b59.png)
(2)如图2,将线段CP绕点C顺时针旋转120°得线段CQ,连接BQ交AC于点D,
①在图中找一个与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/217d37ca5469a57cb7417a2ac0d58efe.png)
②探究BP、CD、BC的数里关系,并说明理由.
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【推荐3】(1)如图1,锐角△ABC中,分别以AB、AC为边向外作等边△ABE和等边△ACD,连接BD,CE,直接写出一对全等三角形:____________________
试猜想BD与CE的大小关系,直接写出结论:________________________
【深入探究】
(2)如图2,△ABC中,∠ABC=45°,AB=10cm,BC=6cm,分别以AB、AC为边向外作正方形ABNE和正方形ACMD,连接BD,求BD的长.
(3)如图3,在(2)的条件下,以AC为直角边在线段AC的左侧作等腰直角△ACD,直接写出BD的长是_________________________.
试猜想BD与CE的大小关系,直接写出结论:________________________
【深入探究】
(2)如图2,△ABC中,∠ABC=45°,AB=10cm,BC=6cm,分别以AB、AC为边向外作正方形ABNE和正方形ACMD,连接BD,求BD的长.
(3)如图3,在(2)的条件下,以AC为直角边在线段AC的左侧作等腰直角△ACD,直接写出BD的长是_________________________.
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解答题-问答题
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适中
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【推荐1】如图,在
中,
,
为
的中点,
,
.
(1)求
的长;
(2)求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f08273d339dc5ddbb89aa67bb8205e6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d70dc2c20619a4fc12a0cfda59af5b69.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/7/ed33463c-8b36-4c44-a762-c996f9fca8d7.png?resizew=172)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b55bc00b3347bf32437887000fe3be66.png)
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【推荐2】如图,在矩形ABCD中,AB=4,BC=3.将△ACD沿对角线AC翻折得到△ACD′,CD′交AB于点F.
(1)判断△ACF的形状,并证明;
(2)直接写出线段AF的长.
(1)判断△ACF的形状,并证明;
(2)直接写出线段AF的长.
![](https://img.xkw.com/dksih/QBM/2021/1/11/2633926803554304/2639317451743232/STEM/86c73a3eb57d432ebcbed460b0dfff18.png?resizew=142)
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解答题-作图题
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【推荐3】问题提出:如果一个多边形的各个顶点均在另一个多边形的边上,则称这个多边形为另一多边形的内接多边形
问题探究:
![](https://img.xkw.com/dksih/QBM/2019/1/11/2116473922117632/2118545674682368/STEM/6bbc207aea5b4c7ca95e6288ff3f62ac.png?resizew=458)
(1)如图1,正方形PEFG的顶点E、F在等边三角形ABC的边AB上,顶点P在AC边上.请在等边三角形ABC内部,以A为位似中心,作出正方形PEFG的位似正方形P'E'F'G',且使正方形P'E'F'G'的面积最大(不写作法)
(2)如图2,在边长为4正方形ABCD中,画出一个面积最大的内接正三角形,并求此最大内接正三角形的面积
拓展应用:
(3)如图3,在边长为4的正方形ABCD中,能不能截下一个面积最大的直角三角形,并使其三边比为3:4:5,若能,请求出此直角三角形的最大面积,若不能,请说明理由.
问题探究:
![](https://img.xkw.com/dksih/QBM/2019/1/11/2116473922117632/2118545674682368/STEM/6bbc207aea5b4c7ca95e6288ff3f62ac.png?resizew=458)
(1)如图1,正方形PEFG的顶点E、F在等边三角形ABC的边AB上,顶点P在AC边上.请在等边三角形ABC内部,以A为位似中心,作出正方形PEFG的位似正方形P'E'F'G',且使正方形P'E'F'G'的面积最大(不写作法)
(2)如图2,在边长为4正方形ABCD中,画出一个面积最大的内接正三角形,并求此最大内接正三角形的面积
拓展应用:
(3)如图3,在边长为4的正方形ABCD中,能不能截下一个面积最大的直角三角形,并使其三边比为3:4:5,若能,请求出此直角三角形的最大面积,若不能,请说明理由.
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