1 . 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)
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2 . 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)
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3 . 下面是小芳同学设计的“过直线外一点作这条直线的垂线”的尺规作图过程.
及直线
外一点P.
求作:直线
的垂线,使它经过点P.
作法:
①以P为圆心,大于P到直线l的距离为半径作弧,交直线l于A,B两点:
②连接
和
;
③作
的角平分线
,交直线l于点Q;
④作直线
.
∴直线
就是所求的直线.
根据小芳设计的尺规作图过程,解答下列问题:
(1)使用直尺和圆规,补全图(保留作图痕迹);
(2)写出证明过程和依据.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
求作:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
作法:
①以P为圆心,大于P到直线l的距离为半径作弧,交直线l于A,B两点:
②连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6e61620a272dada8d4b9a9fab6379dfd.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/30513ea48bc1ef3ae78adac83d894f14.png)
③作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cb686e4f5e3938575bc547e849d5513f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
④作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
∴直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
根据小芳设计的尺规作图过程,解答下列问题:
(1)使用直尺和圆规,补全图(保留作图痕迹);
(2)写出证明过程和依据.
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名校
4 . 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)
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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
5 . 下面是“作三角形一边中线”的尺规作图过程
已知:![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
求作:
边上的中线
.
作法:如图,
为圆心,
长为半径作弧,两弧相交于
点;
(2)作直线
,
与
交于
点,所以线段
就是所求作的中线.
根据上述的作法,
(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: ∵![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18c2fa05f042b6023a570dcea68a473.png)
∴四边形
是平行四边形(① )
∵
与
交于
点
∴
(② )
∴
是
的中线.
已知:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
求作:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
作法:如图,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ab609a6574633ebabcff3e73fa862081.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5881068127a39caf319492b4177204f0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
(2)作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
根据上述的作法,
(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: ∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18c2fa05f042b6023a570dcea68a473.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76981622e3684aca12435743b6a08a8c.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/532aece6cfd67e2a97977eed978dbf2b.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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6 . 下面是小方设计的“作等边三角形”的尺规作图过程.
已知:线段
.
求作:等边三角形
.
作法:如图,
①以点A为圆心,以
的长为半径作
;
②以点B为圆心,以
的长为半径作
,交
于C;
③连接
.
所以
就是所求作的三角形.
(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:∵点B,C在
上,
∴
(_____________)(填推理的依据).
同理∵点A,C在
上,
∴
.
∴______=_______=_______.
∴
是等边三角形.(_____________)(填推理的依据).
已知:线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
求作:等边三角形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
作法:如图,
①以点A为圆心,以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c20d0b44025a639ce3a92d639dae587.png)
②以点B为圆心,以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6f6558fef858bf27e9811c2d9426fe7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c20d0b44025a639ce3a92d639dae587.png)
③连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5345f892125374809bcaaab5da8d3f5f.png)
所以
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
(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:∵点B,C在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3c20d0b44025a639ce3a92d639dae587.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
同理∵点A,C在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c6f6558fef858bf27e9811c2d9426fe7.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3570a95f68349fcd9417fcda62e78e7e.png)
∴______=_______=_______.
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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名校
7 . 乐乐发现,任意一个直角三角形都可以分割成两个等腰三角形,已知:在中,
.
求作:直线,使得直线
将
分割成两个等腰三角形.
下面是乐乐设计的尺规作图过程.
作法:如图,①作直角边的垂直平分线
,与斜边
相交于点
;
②作直线. 所以直线
就是所求作的直线.
根据乐乐设计的尺规作图过程,解决下列问题:
(1)使用直尺和圆规,补全图形(保留作图痕迹);
(2)证明直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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名校
8 . 尺规作图:过直线外一点作这条直线的平行线.
已知:如图,直线
和直线
外一点
.
求作:直线
,使得
,且
经过点
.
①在直线
上任取一点
,以点
为圆心,任意长为半径作弧,交
于点
;
②连接
,分别以
为圆心,大于
长为半径作弧,两弧交于
,
两点;
③作直线
,交
于点
;
④作射线
,在线段
的延长线上取点
,使得
;
⑤作直线
,则
即为所求作直线
.
(2)完成下面的证明.
证明:连接
,
,
∵
是线段
的垂直平分线,垂足为
,
∴
.
又∵
,
∴四边形
为( )(用汉字填四边形名称)
(_____________________)(填推理依据).
∴
(___________________)(填推理依据).
即
.
已知:如图,直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
求作:直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3969a47550d3622608f5b868e6d7d7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
①在直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0f85fca60a11e1af2bf50138d0e3fe62.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
②连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/098a3e7d1f1890863b7483a98b618119.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c93148adbc6e856da9a9d263f485d003.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
③作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
④作射线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/aaf1438142deeac876fc7dc50552e552.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c940869dd66321d1132828a95cb5353.png)
⑤作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/294f5ba74cdf695fc9a8a8e52f421328.png)
(2)完成下面的证明.
证明:连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0944865f14b13a772526c4cae4f29c99.png)
又∵
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4c940869dd66321d1132828a95cb5353.png)
∴四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
(_____________________)(填推理依据).
∴
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4adf90a8c2b29334cdc5aa5b554991f9.png)
即
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a3969a47550d3622608f5b868e6d7d7.png)
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9 . 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)
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10 . 如图,已知线段
.求作:
的垂线,使它经过点A.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/9716f141-c6f4-4593-b538-8c790b964650.png?resizew=118)
下面是小军设计的“过线段端点作这条线段的垂线”的尺规作图过程.
作法:①以点A为圆心,
长为半径作弧,交线段
的延长线于点C;
②分别以点B和点C为圆心,大于
的长为半径作弧,两弧相交于直线BC上方的点D;
③作直线
.所以直线
就是所求作的垂线.
根据小军设计的尺规作图过程,
(1)使用直尺和圆规,补全图形;(保留作图痕迹)
(2)证明这种作法的正确性(即求证
).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/1/5/9716f141-c6f4-4593-b538-8c790b964650.png?resizew=118)
下面是小军设计的“过线段端点作这条线段的垂线”的尺规作图过程.
作法:①以点A为圆心,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7dea2ae9d515f9ab351ad72306b776ee.png)
②分别以点B和点C为圆心,大于
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8d452f74791f58e8400cb8d2d6038dc4.png)
③作直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
根据小军设计的尺规作图过程,
(1)使用直尺和圆规,补全图形;(保留作图痕迹)
(2)证明这种作法的正确性(即求证
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9060f03b9ee41d70d135b1e1a8902ce9.png)
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