名校
1 . 小明在学习了圆内接四边形的性质“圆内接四边形的对角互补”后,想探究它的逆命题“对角互补的四边形的四个顶点在同一个圆上”是否成立.他先根据命题画出图形,并用符号表示已知,求证.
已知:如图,在四边形
中,
.
求证:点
在同一个圆上.
他的基本思路是依据“不在同一直线上的三个点确定一个圆”,先作出一个过三个顶点
的
,再证明第四个顶点
也在
上.
具体过程如下:
步骤一 作出过
三点的
.
如图1,分别作出线段
的垂直平分线
,
设它们的交点为
,以
为圆心,
的长为半径作
.
连接
,
(①______).(填推理依据)
.
点
在
上.
步骤二 用反证法证明点
也在
上.
假设点
不在
上,则点
在
内或
外.
ⅰ.如图2,假设点
在
内.
延长
交
于点
,连接
.
(②______).(填推理依据)
是
的外角,
(③______).(填推理依据)
.
.
这与已知条件
矛盾.
假设不成立.即点
不在
内.
ⅱ.如图3,假设点
在
外.
设
与
交于点
,连接
.
.
是
的外角,
.
.
.
这与已知条件
矛盾.
假设不成立.即点
不在
外.
综上所述,点
在
上.
点
在同一个圆上.
阅读上述材料,并解答问题:
(1)根据步骤一,补全图1(要求:尺规作图,保留作图痕迹);
(2)填推理依据:①______,②______,③______.
已知:如图,在四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd602a957d7f6d0940f79a1121b78c6.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/16/c47d4791-3f7d-498a-ac80-1ac7ddf2393a.png?resizew=132)
求证:点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
他的基本思路是依据“不在同一直线上的三个点确定一个圆”,先作出一个过三个顶点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
具体过程如下:
步骤一 作出过
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
如图1,分别作出线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/374fe9986ebbc986fc422e514ab93a51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/280860dd039e1305a5ccc455f63e8223.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/16/93a582e4-1496-4432-9f4d-0da9602ef262.png?resizew=161)
设它们的交点为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/84c83984c62d390c6b30efa5d4e560de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bf8109c7ef3b5f448187a18230381ff2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b1ffe501e7fd2693aee473b426fd4fe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
步骤二 用反证法证明点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
假设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
ⅰ.如图2,假设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/16/4fbff140-e8a9-4d5f-8e33-8e6dccc33b37.png?resizew=143)
延长
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6795cae2df43a722e1355e9562d93c09.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83c09eec4e14a861af83d7828797d176.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d08df5c56bc4b5e8c36b0f42d53ec640.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/583061d83e4013cf3d4e51af1d6ad6d1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ca4de204c6d5e4b7bfb00cfceb445e5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5088b8cbc755b1d6b78e5185987b0796.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e98fccf0bb5849463bb3c7e69f30198.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/803c0a6e500e007441b642cbe3c9cda3.png)
这与已知条件
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd602a957d7f6d0940f79a1121b78c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
ⅱ.如图3,假设点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/16/08b8f3d0-9ffb-4bb4-bfa0-daa892697c5c.png?resizew=137)
设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c296e45b84cf67a98939aa7334e7d478.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4999f12484db6bb05f2b22ab76312e72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39b04e23ad5e1a1be625f73679c1a250.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b82e47950865fba90dbc5b31e21e928.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/953f36b59a6bec2e18de31c6da1b567c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a20e1a95fa95baa2911f30b88fb005d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/692bb8ba82ac5900b75a069d25c6fecb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/549aa6b3dc272af85c4a5fca6e4986b8.png)
这与已知条件
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8bd602a957d7f6d0940f79a1121b78c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
综上所述,点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0c82a10b4f0c9323d726804c89dd9548.png)
阅读上述材料,并解答问题:
(1)根据步骤一,补全图1(要求:尺规作图,保留作图痕迹);
(2)填推理依据:①______,②______,③______.
您最近一年使用:0次
2024-01-13更新
|
205次组卷
|
2卷引用:北京市朝阳区2023-2024学年九年级上学期期末数学试题
名校
2 . 尺规作图之旅:如图
是一副纯手绘的画作,其中用到的主要工具就是直尺和圆规,在数学中,我们也能通过尺规作图创造出许多带有美感的图形.尺规作图起源于古希腊的数学课题,只允许使用圆规和直尺,来解决平面几何作图问题.
而这些尺规作图的背后都与我们学习的数学原理密切相关,下面是用尺规作一个角等于已知角的方法及说理,请补全过程.
已知:如图
,
,
使
.
作法:①如图,以
为圆心,以______为半径画弧,分别交
,
于点
,
;
②画一条射线
,以点
为圆心,以______长为半径画弧,交
于点
;
③以______为圆心,以______长为半径画弧,交前面的弧于点
,
④作射线
,
就是所求作的角.
(2)如图
,
,过点
画射线
,则
,
,
,
,
求证:
,
证明:
,
≌
.(______)
所以
.(______)
(3)请按照上面的范例,完成尺规作图:
①在图
中画出
的角平分线;
②在图
中直线上找到一点
,使它到点A,点
的距离相等.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bdaa19de263700a15fcf213d64a8cd57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07096af3b99fd1cb11c31f19a2c6408e.png)
已知:如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07e3d51307daa0cfab77eec9311d91bf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27949b9eac364001ddbd7ec0ef2e0142.png)
作法:①如图,以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b90e0f35eda1a729fed485f83da5ea9d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
②画一条射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b48e33eaa40d01f976230d4b73ae91dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/12fe32dfbd66709875c5b9f79c9496da.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b48e33eaa40d01f976230d4b73ae91dc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/713877f832ddbfc9d52bd9902cda13b4.png)
③以______为圆心,以______长为半径画弧,交前面的弧于点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
④作射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6637859805b7b01f7a9a64771563406e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/07e3d51307daa0cfab77eec9311d91bf.png)
(2)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b8860d9787671b53b1ab68b3d526f5ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5b3bd5e6bc2a0a277d279bb01af9584.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6637859805b7b01f7a9a64771563406e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27949b9eac364001ddbd7ec0ef2e0142.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9ec841221c64807bdb8ce6fbac6ed013.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3d1b5e4a70bb62763b35bb7cbd01912.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/063be481ae445e25b5b98fb3870efd16.png)
求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27949b9eac364001ddbd7ec0ef2e0142.png)
证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30c1198e889b163eb77140a0915196ca.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0fca1a15b67c2375e838fa0937748e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c2b4890319264aa23ad9b059d05b458.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/27949b9eac364001ddbd7ec0ef2e0142.png)
(3)请按照上面的范例,完成尺规作图:
①在图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/61128ab996360a038e6e64d82fcba004.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d7b2fe01a33c4825f9974ed9663a99c.png)
②在图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d91e07104b699c4012be2d26160976a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/9/26/23ad7ff3-c73e-4bcf-afdd-cbcd2c41e5de.jpg?resizew=136)
您最近一年使用:0次
2023-08-24更新
|
85次组卷
|
3卷引用:河南省驻马店市驿城区第二初级中学2022-2023学年七年级下学期期末数学试题
3 . 数学课上,王老师布置如下任务:如图,已知
,点
是射线
上的一个定点,在射线
上求作点
在
和
之间),使
.
下面是小路设计的尺规作图过程.
作法:作线段
的垂直平分线l,直线l交射线
于点C,则点C即为所求.
根据小路设计的尺规作图过程,
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/27/a9948107-753a-4ffc-abea-60d2401c1613.png?resizew=202)
(1)使用直尺和圆规,补全图形;(保留作图痕迹)
(2)完成下面的证明:
证明:连接
,
∵直线l为线段
的垂直平分线,
∴
,( )(填推理的依据)
∴
,
∴
( )(填推理的依据)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c45c354f194d9410d119c5054548de93.png)
(3)能否在射线
上再求作点
,使
.若能简要说明作法,并使用直尺和圆规画出图形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39da1b9bc3e46a4f9317c393978bfb4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d50703c46b6153945d718b198f03b4b5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f2801d22074121bca2fcf0cd2531ef2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54a5d7d3b6b63fe5c24c3907b7a8eaa3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c45c354f194d9410d119c5054548de93.png)
下面是小路设计的尺规作图过程.
作法:作线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
根据小路设计的尺规作图过程,
![](https://img.xkw.com/dksih/QBM/editorImg/2023/3/27/a9948107-753a-4ffc-abea-60d2401c1613.png?resizew=202)
(1)使用直尺和圆规,补全图形;(保留作图痕迹)
(2)完成下面的证明:
证明:连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
∵直线l为线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ac2e854df9867fea0ace9156bc215da.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14b1848fa4f378761a9d734e9d0aa9de.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3a77cc08ff91bb2f732109d8a42ee365.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c45c354f194d9410d119c5054548de93.png)
(3)能否在射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f50b3ae183997b707d16eb4e7f6712fa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1d6c3cb56e4fcc971761655bc401ec7b.png)
您最近一年使用:0次
名校
4 . 如图,
是
的直径,过点A作
的切线
,点P是射线
上的动点,连接
,过点B作
,交
于点D,连接
.
(2)证明:
是
的切线.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/465cc28d412584e98f57a8745e98b668.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e0629ce42392a7fe9be21d25c39c3e64.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
您最近一年使用:0次
名校
5 . 如图,四边形
是平行四边形.
的角平分线
,交
于点E;在线段
上截取.
,连接
;(不写作法,保留作图痕迹)
(2)在(2)所作图中,证明四边形
是菱形.请完成下面的填空,补全证明过程.
证明:∵四边形
是平行四边形,
∴ ①
∴
,
∵
平分
,
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06bb6eb5554f82a9031eb649a212af72.png)
∴
②
∴
③
,
,
又∵
,
∴四边形
是 ④ ,
∵
,
∴四边形
是菱形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b6961ff0c507e6c21ee3af44e887c8c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(2)在(2)所作图中,证明四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/369eb8ad56da7dc1cdb7c43762be4bee.png)
证明:∵四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
∴ ①
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fc8289b804775f99d4e883697ed90c3a.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/06bb6eb5554f82a9031eb649a212af72.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d63783f365e666ace307cadcaba60fa.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c26a46e7879436d532af3f4b6e258a81.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b4d998ee9f627fccb48171d00e44c8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/475b4502140ea9bcdda0c9c0eff6481f.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5bf7dc43a39738e3e2a0b819be505c6e.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/369eb8ad56da7dc1cdb7c43762be4bee.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c1cba730226f8996e86127ff643ce63.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/369eb8ad56da7dc1cdb7c43762be4bee.png)
您最近一年使用:0次
6 . 如图,在菱形
中,对角线
、
相交于点
.
(1)尺规作图:在
的延长线上截取
,连接
,再过点
作
的垂线交
于点
(保留作图痕迹,不写作法);
为矩形.(补全证明过程)
证明:
四边形
是菱形
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfc1f76257275ab4b04f9bc913535670.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec10e2a03f40f90a79f2d4711f83de4.png)
① ,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b2fb91028cda3644b82b04da29cbe52.png)
为
的中位线
②
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d217d4275d6cb2f4b5d74d02059c70d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddd0101bc314b14516b2e824aeaa2245.png)
③
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ed2d95f956bf43f94a5cf963e55943.png)
四边形
为矩形.( ④ )
进一步研究上述问题发现,当
和
满足位置关系: ⑤ 时,四边形
为正方形.
![](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/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(1)尺规作图:在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6b1bd1adfe4cc6566218f19970c2fd3b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df1ac3ab8d3ce5e61fd2f89761f80976.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b47e44229585c09a3eec68277369beb8.png)
证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/798857f4ba597343bb9e09e14cbb248a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cfc1f76257275ab4b04f9bc913535670.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2ec10e2a03f40f90a79f2d4711f83de4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7b2fb91028cda3644b82b04da29cbe52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f78ab2a7a8fc8fbfb59031dd46a6daa8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d217d4275d6cb2f4b5d74d02059c70d8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddd0101bc314b14516b2e824aeaa2245.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20ed2d95f956bf43f94a5cf963e55943.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b47e44229585c09a3eec68277369beb8.png)
进一步研究上述问题发现,当
![](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/b47e44229585c09a3eec68277369beb8.png)
您最近一年使用:0次
7 . 如图,在四边形
中,
,
.
上截取
,连接
,作
的角平分线
,分别交
于点F、G,连接
.(保留作图痕迹,不写作法)
(2)在(1)所作图形中,求证:
.(请补全下面的证明过程,不写证明理由)
证明:∵
是
的角平分线,
∴ ,
∵
,
∴ ,
∴
,
∴ ,
又∵
,
∴ ,
∴四边形
是平行四边形,
∴
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dceb5cc71fc50f20649f6b9535fd914.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a626eb07074f222d52c129b275e173d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be5fc6ead6416492c231c320a5486f86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2947ca8e0cdbeb4aab80ce9e7b63ba98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ab6fb62038b9207669ce6f8309fb142.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/31e55e398e8520d8a36fb5a625a085b8.png)
(2)在(1)所作图形中,求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d440215e4ca391884e61b1017e329e4.png)
证明:∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/77a7e4a6765ce78b05ee97764771e01f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2947ca8e0cdbeb4aab80ce9e7b63ba98.png)
∴ ,
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6dceb5cc71fc50f20649f6b9535fd914.png)
∴ ,
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2a40e29b48b7a23bc25eb615bbe5f5d.png)
∴ ,
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/be5fc6ead6416492c231c320a5486f86.png)
∴ ,
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7419886f571571b57b61a6a8980305e8.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4d440215e4ca391884e61b1017e329e4.png)
您最近一年使用:0次
8 . 学习完四边形的知识后,小明想出了“作三角形一边上的中线”的另一种尺规作图的方法,下面是具体过程.
已知:
(如图).
求作:
边上的中线
.
作法:
①以点A为圆心,
长为半径画弧,以点C为圆心,
长为半径画弧,两弧相交于点 P;
②作射线
,与
交于点D,线段
就是所求作的中线.
(2)求证:
是
边上的中线.
已知:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
求作:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
作法:
①以点A为圆心,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
②作射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
您最近一年使用:0次
9 . 已知四边形
是平行四边形,
.
的平分线交
于点E,在
上截取
,连接
;(要求保留作图痕迹,不写作法)
(2)求证:四边形
是菱形.(补全下列证明过程)
证明:
四边形
为平行四边形,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
,
___________.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
平分
,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
,
___________.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
,
又![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
,
___________.
又![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
,
四边形
为平行四边形,
又
___________.
四边形
是菱形.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3039c774e00f6520449aa9d4b3c45464.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2947ca8e0cdbeb4aab80ce9e7b63ba98.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/14833dbeed409b33acd4c9071fd0be36.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
(2)求证:四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac37366d2b54dc7d9a95ac6ddda5f3a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22b5cdb22521df6390796ab18339e895.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bae74865b97db7ced7fbf67303668ee.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c7e48723871f06a6aeae31a2a1ff79d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
又
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/16f3d198e76391779fa3badc848c8ac8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2dde327febef2331a4766a79b433cc02.png)
您最近一年使用:0次
10 . 如图,
中,
.
.
作法:
①作线段
的垂直平分线
交
于点
;
②连接
并延长,在延长线上截取
;
③连接
.
则四边形
为所求作的矩形.
(1)使用直尺和圆规,依作法补全尺规作图(保留作图痕迹);
证明:
是线段
的垂直平分线,
,
,
四边形
为平行四边形(______)(填推理依据).
,
平行四边形
为矩形(______)(填推理依据).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dd967903ed5a6f640a5b801ec8be0070.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/45acdbac251ca6b76a166c1242e71df9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
作法:
①作线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
②连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5fce05f91ce17863abc8a5a8f6002b75.png)
③连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2cd472806caed4b14ceee23f4c9a04f.png)
则四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(1)使用直尺和圆规,依作法补全尺规作图(保留作图痕迹);
证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/26efaff579cd0b6151c68225fafb5720.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb7475a34bbc1cd1e1b468056d087828.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d3319f5fb4ce21a2893a9d4f340de2cd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5d6808b55943e0ab815da922bf1b095e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2de0d10ef8b748d4531250c37c5d3f9e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
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