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 . 数学活动课上,小明同学根据学习函数的经验,对函数的图象、性质进行了探究.如图1,已知在
中,
,
,
,点P为AB边上的一个动点,连接PC,设
,
,
![](https://img.xkw.com/dksih/QBM/2022/1/16/2895927304634368/2934571169914880/STEM/184fab3d-5b3c-4012-90b8-5cb1a686684a.png?resizew=317)
(1)当
时,则 x= ;y= ;
(2)填表:
(说明:补全表格时相关数值保留一位小数)(参考数据:
;
).
(3)试求y与x之间的函数关系式;
a、建立平面直角坐标系,如图2,描出剩余的点,并用光滑的曲线画出该函数的图象;
b、结合画出的函数图象,写出该函数的两条性质:
① ;
② .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f8f88798ec42a58dccd212586382b23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a355958abf7dc0f2eb949584cb87907b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44f4bcb7ddcdbb66f0304d0531e84c6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/40eee6f84b2af5e06da1cd3d0a1f3a0f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/daa03b06bc3ffc95899645c08b21fcd1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c2a80b46e3ef7314b35df0517c969608.png)
![](https://img.xkw.com/dksih/QBM/2022/1/16/2895927304634368/2934571169914880/STEM/184fab3d-5b3c-4012-90b8-5cb1a686684a.png?resizew=317)
(1)当
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/df2e118d8156830746055c1b2e759ab0.png)
(2)填表:
x/cm | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 |
y/cm | 2 | 1.8 | 1.7 | 2 | 2.3 | 2.6 | 3 |
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/47bb3f35e3db7c1f3a3dd3eb20151b5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f4f0684b92d6d24d90a6fb39d3d6529d.png)
(3)试求y与x之间的函数关系式;
a、建立平面直角坐标系,如图2,描出剩余的点,并用光滑的曲线画出该函数的图象;
b、结合画出的函数图象,写出该函数的两条性质:
① ;
② .
您最近一年使用:0次
2022-03-12更新
|
213次组卷
|
2卷引用:江西省南昌市财大附中2021-2022学年九年级上学期期末联考数学试题
3 . 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
4 . 下面是某学习小组设计的“过圆外一点作圆的切线”的尺规作图过程.
已知:
及圆外一点P.
求作:过点P且与
相切的直线.
作法:如图,①连接
,分别以O,P为圆心,大于![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
长为半径画弧,两弧交于M,N两点;②作直线
,与
交于点Q,以Q为圆心,以
长为半径作圆,交
于A,B两点;③作直线
,
.则直线
,
是所求作的
的切线.
根据该小组设计的尺规作图过程:
(1)使用直尺和圆规,按照上述作法补全图形;(保留作图痕迹)
(2)完成下面的证明.
,
,
,
,
,
∵
,
,
∴
是
的垂直平分线,( )(填推理的依据)
∴Q为
中点,
,
∴
为
的直径,
∴
,( )(填推理的依据)
∵A点在
上,
∴
是
的切线.( )(填推理的依据)
已知:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
求作:过点P且与
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
作法:如图,①连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f89eef3148f2d4d09379767b4af69132.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9f0009063fe00277645aff1be6e32471.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
根据该小组设计的尺规作图过程:
(1)使用直尺和圆规,按照上述作法补全图形;(保留作图痕迹)
(2)完成下面的证明.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/438f34bc8b04e8c494b91306ac6fe352.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a4cfef623a9534b5708df5f95f1760a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5845ffc7d1af9053b027cf9e726f5367.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f5dbfcff5c8b5d4e312a9247b2b8b0a4.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1ccebeed9252beaea560ab6697cacd76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/212b4f502299f95b58bde72ad9b59023.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
∴Q为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b4b389c6e8086c68c36313cb620c1078.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb52bba9798c625c7cd778636bceea32.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3b15b185062675704cf6a41d5f16b232.png)
∵A点在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd33764ff4efddfe11a98a609753715c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
您最近一年使用:0次
5 . 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
6 . 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
名校
7 . 如图,在
中,
的角平分线
交
于点D,并在射线
上另取一点E(不与A重合),使得
,连接
;(保留作图痕迹,不写作法)
(2)在(1)所作图形中,若D恰为线段
的中点,求证;
.(请补全下面的证明过程,不写证明理由)
证明:∵D为
中点
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532aece6cfd67e2a97977eed978dbf2b.png)
∴在
和
中
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6ad3e834b13b8cd382b406612d5f4a2.png)
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/038df92742be9dca6e3d62a262c6893e.png)
∴
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102ebb908833439c37e364f60f1d4ebb.png)
又∵
是
的角平分线
∴②
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52154425097228893a63b790e1c20d9.png)
∴③
又∵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b59f0e6f7fa9c50e7f5cc146ba1af1.png)
∴![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
由此发现一个结论,请完成下列命题:
如果一个三角形的一个内角的角平分线又是对边上的中线,那么④ .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.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/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/518586d91b63569fc317b323835a0c2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
(2)在(1)所作图形中,若D恰为线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
证明:∵D为
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532aece6cfd67e2a97977eed978dbf2b.png)
∴在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab2a2834d80ff574e79eae8ca8d4e94f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b2b265d121f9ebc13671a5719604476a.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e6ad3e834b13b8cd382b406612d5f4a2.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/038df92742be9dca6e3d62a262c6893e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17101383eb0787edaaa35adfcd20d5c9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/102ebb908833439c37e364f60f1d4ebb.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
∴②
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e52154425097228893a63b790e1c20d9.png)
∴③
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5b59f0e6f7fa9c50e7f5cc146ba1af1.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
由此发现一个结论,请完成下列命题:
如果一个三角形的一个内角的角平分线又是对边上的中线,那么④ .
您最近一年使用:0次
名校
8 . 如图,在
中,
,
,作线段
的垂直平分线,交
于点D,交
于点E.
(2)求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d3c9ce32b721995f355eea411340e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f9811b144b053df12261e59bdaf53847.png)
您最近一年使用:0次
2024-03-31更新
|
234次组卷
|
5卷引用:2024年河南省安阳市安阳县中招模拟第一次联考数学模拟预测题
名校
9 . 已知四边形
为正方形,点
在
边上,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/27/5304c09c-d09f-4b43-a038-0f42d4ac1650.png?resizew=113)
(1)尺规作图:过点
作
于点
,交
于点
(保留作图痕迹,不写作法,不下结论);
(2)求证:
.(请补全下面的证明过程)
证明:∵正方形
,
∴
,![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6217ba54baf35aca59d9b89ded7367e.png)
________
,
∴
,
∵
,
∴
,
∴
,
∴
________,
在
与
中
,
( )里填________
∴
(
),
∴
.
通过上面的操作,进一步探究得到这样的结论:两端点在正方形的一组对边上且
______的线段长相等.
![](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/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/27/5304c09c-d09f-4b43-a038-0f42d4ac1650.png?resizew=113)
(1)尺规作图:过点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb91cb9a5a14169845d700fbd95890ac.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/a0ed1ec316bc54c37c4286c208f55667.png)
(2)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04f8eebda19eded2b059774a8c2666c3.png)
证明:∵正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3570a95f68349fcd9417fcda62e78e7e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d6217ba54baf35aca59d9b89ded7367e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e1c9ae241fd78126274c65e17990c88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b0c06422e1d55db3077257af113df4bb.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e4500c083ce75a99fd640c3067e600d9.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fb91cb9a5a14169845d700fbd95890ac.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/17d2242e660785814aa933b93ea28a0e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fdf65f2f3f02118a8956c76ba9f55fae.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5c650fe55b7603f106c53ca2423451c6.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e742966e3711cfa53dce04022acf4bcc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6830ebecddbd9759be626289c408e4f3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62d1f23c643b53965a5beb8200354397.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9b3be9504b4d7f24528377970a5aa198.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/922514832587986f60a0a3fa4ec3ca0c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4351a730f61bb998bab8f0b7848912d7.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04f8eebda19eded2b059774a8c2666c3.png)
通过上面的操作,进一步探究得到这样的结论:两端点在正方形的一组对边上且
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7d8b8edd94bc4d5d517ec77e56800e41.png)
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名校
10 . 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)
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