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
2 . 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
3 . 尺规作图:
已知:如图1,直线MN和直线MN外一点P.
求作:直线PQ,使直线PQ
MN.
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/d33e57b8-4b96-4fe6-89b9-35cda87df754.png?resizew=263)
小智的作图思路如下:
①如何得到两条直线平行?
小智想到,自己学习线与角的时候,有4个定理可以证明两条直线平行,其中有“内错角相等,两条直线平行”.
②如何得到两个角相等?
小智先回顾了线与角的内容,找到了几个定理和1个概念,可以得到两个角相等.小智又回顾了三角形的知识,也发现了几个可以证明两个角相等的定理.最后,小智选择了角平分线的概念和“等边对等角”.
③画出示意图:
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/30bb1074-29f7-45f5-b9d9-3d7fd90b2401.png?resizew=421)
④根据示意图,确定作图顺序.
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/99266139-5b27-4fae-86da-de956d5d841b.png?resizew=542)
(1)使用直尺和圆规,按照小智的作图思路补全图形1(保留作图痕迹);
(2)完成下面的证明:
证明:∵AB平分∠PAN,
∴∠PAB=∠NAB.
∵PA =PQ,
∴∠PAB=∠PQA ( ① ).
∴∠NAB =∠PQA.
∴PQ
MN ( ② ).
(3)参考小智的作图思路和流程,另外设计一种作法,利用直尺和圆规在图2中完成.(温馨提示:保留作图痕迹,不用写作法和证明)
已知:如图1,直线MN和直线MN外一点P.
求作:直线PQ,使直线PQ
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a9bfa68259d7a331be323b2038d628a.png)
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/d33e57b8-4b96-4fe6-89b9-35cda87df754.png?resizew=263)
小智的作图思路如下:
①如何得到两条直线平行?
小智想到,自己学习线与角的时候,有4个定理可以证明两条直线平行,其中有“内错角相等,两条直线平行”.
②如何得到两个角相等?
小智先回顾了线与角的内容,找到了几个定理和1个概念,可以得到两个角相等.小智又回顾了三角形的知识,也发现了几个可以证明两个角相等的定理.最后,小智选择了角平分线的概念和“等边对等角”.
③画出示意图:
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/30bb1074-29f7-45f5-b9d9-3d7fd90b2401.png?resizew=421)
④根据示意图,确定作图顺序.
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/99266139-5b27-4fae-86da-de956d5d841b.png?resizew=542)
(1)使用直尺和圆规,按照小智的作图思路补全图形1(保留作图痕迹);
(2)完成下面的证明:
证明:∵AB平分∠PAN,
∴∠PAB=∠NAB.
∵PA =PQ,
∴∠PAB=∠PQA ( ① ).
∴∠NAB =∠PQA.
∴PQ
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a9bfa68259d7a331be323b2038d628a.png)
(3)参考小智的作图思路和流程,另外设计一种作法,利用直尺和圆规在图2中完成.(温馨提示:保留作图痕迹,不用写作法和证明)
![](https://img.xkw.com/dksih/QBM/2022/1/14/2894320983465984/2897283834994688/STEM/62c71d46-05a8-4dd7-9f42-3e23f77647e9.png?resizew=247)
您最近一年使用:0次
名校
4 . 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
5 . 在学完圆的相关知识后,小东设计了一个“过直线外一点作该直线的垂线”的方法.
下面是小东设计的尺规作图过程.
已知:如图,直线
和直线外一点
.
求作:直线,使得
.
作法:如图,
在直线
上取一点
点
在点
的左侧
,连接
;
作线段
的垂直平分线
,交
于点
;
以点
为圆心,
长为半径作圆,交直线
于点
;
作直线
,则直线
即为所求作的直线.
根据小东设计的尺规作图过程,完成下列问题:
(1)使用直尺和圆规,补全图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d47765a4ef6188f533afeaf67d448257.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/04582116cd765fcc5a52f44279ad6c94.png)
(2)完成下面的证明.
证明:为
的直径,
______
______
填推理的依据
.
直线
即为所求作的直线.
(3)如图
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5ca7d1107389675d32b56ec097464c14.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b143e42eff8c72d4fe319eb120a02284.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411461db15ee8086332c531e086c40c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
您最近一年使用:0次
6 . 如图,在
中,
,
的垂直平分线交
于点
,交
于点
,
的平分线交
于点
,两线交点为点
.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/11/34fe03c2-6165-49e0-b626-573ebc6edb7c.png?resizew=166)
(1)依题意补全图形(要求:尺规作图,保留作图痕迹,不写作法);
(2)连接
,若
,
的周长是
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/998329f9cdb86f5d60d7d5d70fc3781e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.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/dad2a36927223bd70f426ba06aea4b45.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/11/34fe03c2-6165-49e0-b626-573ebc6edb7c.png?resizew=166)
(1)依题意补全图形(要求:尺规作图,保留作图痕迹,不写作法);
(2)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a2fae188b381c5ba01b3d9d742c687dd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6818a98204f62c1b16699d26ca0c3f62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9822e15763be5cb6c49936df274a6748.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
您最近一年使用:0次
7 . 小明在学习《直角三角形的性质》的过程中产生了一个猜想:“在直角三角形中,
角所对的直角边是斜边的一半.”并进行了如下的探究,请完善小明的探究过程.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/18/1d267ffa-a695-472c-8a7d-a70abbaa94d2.png?resizew=170)
(1)结合图形,将小明猜想的命题写成已知、求证:
已知:________________________________________.
求证:
.
(2)补全上述猜想的证明过程.
证明:作线段
的垂直平分线
,交
于点D,交
于点E,连接
.(在图中用尺规作图,并保留作图痕迹)
∵直线
是线段
的垂直平分线,
∴
.(________________________________)(填推理依据).
∴
.(________________________________)(填推理依据).
∵
,
∴
.
∵
中,
,
,
∴
.(________________________________)(填推理依据).
∴
.
∴
.
∵
,
,
∴
.
在
和
中,
,
∴
(________________________________)(填推理依据).
∴
,
∵直线DE是线段AB的垂直平分线,
∴
________.
∴
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/18/1d267ffa-a695-472c-8a7d-a70abbaa94d2.png?resizew=170)
(1)结合图形,将小明猜想的命题写成已知、求证:
已知:________________________________________.
求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd1fe7660ee42b8d3a9c076b700ae59a.png)
(2)补全上述猜想的证明过程.
证明:作线段
![](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/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/68a83fdd2ba72a2dba0b6b10bb3e06b9.png)
∵直线
![](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/7c45087cde2d66377517a3fce5553b35.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7573369ec6e83fdcea41c3de93e5f406.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d3c9ce32b721995f355eea411340e3.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08f57b8380e1c9d059087e2bd121153f.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/56d3c9ce32b721995f355eea411340e3.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/661b25962bb3cfc485e51f99e8e5a945.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13c10daf6b94091ccc96364935bf6f4e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bcb4c8934b3e1ebcd72b901db2464cce.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32c38dfd14dde969702dff97ef2270f9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/79fb061226b947595511ad5a3a994695.png)
在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a25c28359f8d8da9eaf4672a6cf8ae4f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcb49df05f2e31d005735c3f14a21d30.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9efeb07c149df941da2bfde2b2d616d3.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5e4f428b37c40951d638556678210de9.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9fc13fe21e64d9b45614ed43be847904.png)
∵直线DE是线段AB的垂直平分线,
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37d8cdf01dc18ff3b8ecbb90a357a113.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bd1fe7660ee42b8d3a9c076b700ae59a.png)
您最近一年使用:0次
8 . 综合与实践
【动手操作】
数学活动课上,老师让同学们探究用尺规作图作一条直线的平行线.已知:如图1,直线l及直线l外一点A.求作:直线
,使得
.小明同学设计的做法如下:
①在直线l上取两点B、C,连接
,以点B为圆心,小于
的长度为半径作弧,交线段
于点D,交线段
于点E;
②分别以点D和E为圆心,以大于
的长为半径作弧,两弧在
内交于点F,作射线BF;
③以点A为圆心,
的长为半径作弧,交射线
于点P,作直线
.
则直线
平行于直线l.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/0b34d13f-150d-45ea-8faa-d17f035f85dd.png?resizew=435)
(1)根据小明同学设计的尺规作图过程,在图2中补全图形;(要求:尺规作图并保留作图痕迹)
【验证证明】
(2)请证明直线
;
【拓展延伸】
(3)已知:如果两条直线平行,则其中一条直线上任意两点到另外一条直线的距离相等.在图2中连接
,
,请直接写出
与
的面积关系_______;
【应用实践】
(4)某市政府为发展新能源产业,决定在如图3所示的四边形
空地上划出20km2区域用于建设新能源产业发展基地.已知在四边形
中,
,
km,
km.为便于运营管理,某公司向政府提出在线段
上取一点E使得四边形
的面积为20km2,则
______km.
【动手操作】
数学活动课上,老师让同学们探究用尺规作图作一条直线的平行线.已知:如图1,直线l及直线l外一点A.求作:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/881ac4722f272263629875ca901250e4.png)
①在直线l上取两点B、C,连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
②分别以点D和E为圆心,以大于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/65026a6b6402cb599e14be8023a3f6a8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
③以点A为圆心,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
则直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/14/0b34d13f-150d-45ea-8faa-d17f035f85dd.png?resizew=435)
(1)根据小明同学设计的尺规作图过程,在图2中补全图形;(要求:尺规作图并保留作图痕迹)
【验证证明】
(2)请证明直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/881ac4722f272263629875ca901250e4.png)
【拓展延伸】
(3)已知:如果两条直线平行,则其中一条直线上任意两点到另外一条直线的距离相等.在图2中连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/48f3c9abbd78e9a6840ee5f30381daac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c025ee3317be1099b7bf03a11e37ed4.png)
【应用实践】
(4)某市政府为发展新能源产业,决定在如图3所示的四边形
![](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/77b7fa2dc328bf1c85d00a2a27eea745.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/54275b7e571660d0a9e0370fbfe5050b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7788830ed1cb3b9c5988f70f43595f2e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2fa7bbd7831e9ff4f8cffc8889d34f05.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/32217457c4e96e2ef155cf15c1b65d97.png)
您最近一年使用:0次
名校
9 . 下面是小李设计的“过圆外一点作圆的一条切线”的尺规作图的过程.
已知:如图1,
及圆外一点P.
求作:过点P作
的一条切线.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/c579cb70-8eae-4896-b829-488557732639.png?resizew=205)
作法:①连接
;
②作
的垂直平分线,交
于点A;
③以A为圆心,
的长为半径作弧,交
于点B;
④作直线
.
即直线
为所求作的一条切线.
根据上述尺规作图的过程,回答以下问题:
(1)使用直尺和圆规,依作法补全图形(保留作图痕迹);
(2)该作图中,可以得到
______
;依据:____________.
已知:如图1,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
求作:过点P作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/c579cb70-8eae-4896-b829-488557732639.png?resizew=205)
作法:①连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
②作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abd13974aebe38eb2a1d744a01ea5aa5.png)
③以A为圆心,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ef4113c492885ba7c47fe42ac792578f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
④作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
即直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2be49c37e30a3ced0364c3e74d8c687.png)
根据上述尺规作图的过程,回答以下问题:
(1)使用直尺和圆规,依作法补全图形(保留作图痕迹);
(2)该作图中,可以得到
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2f3d1fcc7dfe5820e30c7d8109c36e86.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83873a9d782f2588c5eedbfe73f9bc2f.png)
您最近一年使用:0次
2024-01-13更新
|
148次组卷
|
2卷引用:北京市门头沟区2023-2024学年九年级上学期期末数学试题
10 . 如图,在
中,
,
,作
的角平分线,交
于点D.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/dc2c9037-a76b-48a2-9d10-d9bf910ae249.png?resizew=176)
(1)依题意补全图形(要求:尺规作图,保留作图痕迹,不写作法);
(2)求证
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b57fdd2a3642716fcf5100011eb3ec88.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7e3262fc038bbec5e7c8cc47df08bef7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7cbce11aa19b8bd2bf6ee5a834e005de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/13/dc2c9037-a76b-48a2-9d10-d9bf910ae249.png?resizew=176)
(1)依题意补全图形(要求:尺规作图,保留作图痕迹,不写作法);
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3卷引用:辽宁省大连市金州区2023-2024学年八年级上学期期末数学试题
辽宁省大连市金州区2023-2024学年八年级上学期期末数学试题辽宁省大连市中山区2023-2024学年八年级上学期期末数学试题(已下线)专题1.13 角平分线(知识梳理与考点分类讲解)-2023-2024学年八年级数学下册基础知识专项突破讲与练(北师大版)