如图是由边长为1的小正方形组成的
网格,每个小正方形的顶点叫做格点,
、
、
三个格点都在圆上,点
是此圆内的一个格点,仅用无刻度的直尺在给定网格中完成画图,画图过程用虚线表示.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/10/934cd62c-8b23-493d-a65b-5c373c80f324.png?resizew=120)
(1)画出该圆的圆心
;
(2)画出
上的点
,使
长最小;
(3)画出格点
,使
为
的一条切线,并画出过点
的另一条切线
,切点为
.(只需要画出满足条件的一个点
和一个点
即可)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e788125f02d28749c086590197487f2d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dad2a36927223bd70f426ba06aea4b45.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/2/10/934cd62c-8b23-493d-a65b-5c373c80f324.png?resizew=120)
(1)画出该圆的圆心
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
(2)画出
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/acc290b44635265137fdf13146b6a6d9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
(3)画出格点
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1fc56c77464a17a1e97b568762a3e2c6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
更新时间:2023-02-07 21:54:50
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【推荐1】如图所示,在⊙O上有一点C(C不与A、B重合),在直径AB上有一个动点P(P不与A、B重合).试判断PA、PC、PB的大小关系,并说明理由.
![](https://img.xkw.com/dksih/QBM/2018/3/4/1895116736028672/1895833905340416/STEM/3e0cf2f4-31e5-4984-8d3a-0177c8af12de.png)
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【推荐2】(1)如图①,△PAM是等边三角形,在边PM上取点B(点B不与点P,M重合),连接AB,将线段AB绕点A逆时针旋转60°,得到线段AC,连接BC,MC.
①△MAC可以看作△PAB绕点 逆时针旋转 (度)得到的;
②∠PMC= (度).
(2)如图②,△PAM是等腰三角形,∠PAM=90°,AP=AM=
,在边PM上取点B(点B不与点P,M重合),连接AB,将线段AB绕点A旋转,得到线段AC,旋转角为α,连接PC,BC.
①当α = 90°时,若△PBC的面积为1.5,求PB的长;
②若AB=
,求△PBC面积的最大值(直接写出结果即可).
①△MAC可以看作△PAB绕点 逆时针旋转 (度)得到的;
②∠PMC= (度).
(2)如图②,△PAM是等腰三角形,∠PAM=90°,AP=AM=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
①当α = 90°时,若△PBC的面积为1.5,求PB的长;
②若AB=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2967337e3fcb228dded64ab0c41a17e0.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/8/ad2a637a-ffa7-48a6-8396-3c3e92525435.png?resizew=349)
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【推荐3】阅读材料:如图1,若点P是⊙O外的一点,线段PO交⊙O于点A,则PA长是点P与⊙O上各点之间的最短距离.
证明:延长PO交⊙O于点B,显然PB>PA.
如图2,在⊙O上任取一点C(与点A,B不重合),连结PC,OC.
∵PO<PC+OC,
且PO=PA+OA,OA=OC,
∴PA<PC
∴PA 长是点P与⊙O上各点之间的最短距离.
由此可以得到真命题:圆外一点与圆上各点之间的最短距离是这点到圆心的距离与半径的差.请用上述真命题解决下列问题.
(1)如图3,在Rt△ABC中,∠ACB=90°,AC=BC=2,以BC为直径的半圆交AB于D,P是
上的一个动点,连接AP,则AP长的最小值是 .
(2)如图4,在边长为2的菱形ABCD中,∠A=60°,M是AD边的中点,点N是AB边上一动点,将△AMN沿MN所在的直线翻折得到△A′MN,连接A′C,①求线段A’M的长度; ②求线段A′C长的最小值.
证明:延长PO交⊙O于点B,显然PB>PA.
如图2,在⊙O上任取一点C(与点A,B不重合),连结PC,OC.
∵PO<PC+OC,
且PO=PA+OA,OA=OC,
∴PA<PC
∴PA 长是点P与⊙O上各点之间的最短距离.
由此可以得到真命题:圆外一点与圆上各点之间的最短距离是这点到圆心的距离与半径的差.请用上述真命题解决下列问题.
(1)如图3,在Rt△ABC中,∠ACB=90°,AC=BC=2,以BC为直径的半圆交AB于D,P是
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/83bb3820bab977db734f4335e4fde720.png)
(2)如图4,在边长为2的菱形ABCD中,∠A=60°,M是AD边的中点,点N是AB边上一动点,将△AMN沿MN所在的直线翻折得到△A′MN,连接A′C,①求线段A’M的长度; ②求线段A′C长的最小值.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/3/31/6f34bfca-f372-4468-b53a-6cb8b3403f24.png?resizew=323)
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解答题-计算题
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【推荐1】如图,
是
的直径,
是
的弦,连接
.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/16/219c2ded-8afd-498e-bf07-98c92ad3ea5c.png?resizew=174)
(1)求
的度数;
(2)若
,求
的长.
![](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/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6bac3cb4b0162bb7f27e74f5d0008a04.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/16/219c2ded-8afd-498e-bf07-98c92ad3ea5c.png?resizew=174)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d39b8d91afc34e4a9b0fdbb6bafb9087.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33c99fc553ecb3e2afdb7058201bc642.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
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【推荐2】如图,
是
的直径,
,
都是
上的点,且
平分
,过点
作
的垂线交
的延长线于点
,交
的延长线于点
.
(1)求证:
是
的切线;
(2)若
,
,求
的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9ccea461315a9d05aa0193b937d4bfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9ccea461315a9d05aa0193b937d4bfe.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/88929f4ba0851730d5f941d426b87548.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5b44fa421d3ab01eee0d360a503bd212.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8455657dde27aabe6adb7b188e031c11.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e1a9c6a736e6eac98a676fa3232db5a5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b79dd200766db27fb90d6bd1992cf658.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a0ed1ec316bc54c37c4286c208f55667.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/10/8/f6672b0f-edf1-41b5-9074-676550f7d4da.png?resizew=153)
(1)求证:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0215e13a9fb5574d5194aeb9507a98aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d9ccea461315a9d05aa0193b937d4bfe.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e5e8ca18462a8948f237327b429ef974.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b61ffb2fb597719ab912fcd96a0b11c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b41d4070854edfaa24071137b314cb.png)
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【推荐1】如图,矩形
中,
,
.动点E在
边上,以点E为圆心,以
为半径作弧,点G是弧上一动点.
上,当
与弧相切于点G时,则
的值是 ;
(2)如图2,若
连结
,
,分别取
,
,的中点P、Q,连接
,M为
的中点,则
的最小值为 .
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/305a88d4e0249bd16d48eda01331d2d4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f26ad70d2b3aac8604834d57dfc59bb2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d004d2d115b477ade6af7ddb93db0df8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/274cf35acb4a1748d15c39d15a9bea7b.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c1364213f546b37f8764ddcb59e36ae4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/abf80148409afb32ced0b4f59f1ba709.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1e72b2e1ff83e95df048745322982451.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7a5f1641947153c80b987320885a2b57.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/db54223bb3fc2fe2497213a4d1f94827.png)
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【推荐2】如图,△ABC中,∠C=90°,⊙O分别切AB、BC、AC于D、E、 F,若AD=5cm,BD=3cm,试求出△ABC的面积.
![](https://img.xkw.com/dksih/QBM/2011/12/23/1573315529990144/1573315550650368/STEM/a3e660522af44057af8b0f39edb896b5.png?resizew=207)
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解答题-作图题
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【推荐1】如图,在平面直角坐标系中,已知
,请用无刻度直尺按要求画图.
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/19/e887fb7c-ad0a-4209-b48b-5996bce649f3.jpg?resizew=233)
(1)作
关于y轴对称得到的
,点A,B,C分别对应点
,
,
.
(2)找点P,使得线段
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2023/12/19/e887fb7c-ad0a-4209-b48b-5996bce649f3.jpg?resizew=233)
(1)作
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4310db23fc79936c7182361e652bab1a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a18722354086c42e62334983fc50eb6a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/97c01fdc7bc471af0b264a04aef0823e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b1241216f3c1cb5e73043dd1037f556d.png)
(2)找点P,使得线段
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/144fc51e7f91475874ef059f045dd6da.png)
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解答题-作图题
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【推荐2】图1、图2均是
的正方形网格,每个小正方形的顶点称为格点,小正方形的边长为1,点A、B、C均在格点上.在图1、图2中,只用无刻度的直尺,在给定的网格中按要求画图,不要求写出画法,保留作图痕迹.
的中线
;
(2)在图2中
的边
上找到一点F,使
;
;
(3)
______.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/13a96e9c2e2a15130d1d56a1d0e16b72.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
(2)在图2中
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3deee661336684541870cef3da7dcbd3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51b93087e59c2e08258bd523aa77807e.png)
(3)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/58b9a53e552116848849b76dab92d0ac.png)
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