如图①,直线y=-
x+3与x轴相交于点A,与y轴相交于点B,点C(m,n)是第二象限内一点,以点C为圆心的圆与x轴相切于点E,与直线AB相切于点F.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8b2a698891d42c70b597f0da4f215f09.png)
(1)当四边形OBCE是矩形时,求点C的坐标;
(2)如图②,若⊙C与y轴相切于点D,求⊙C的半径.
17-18九年级下·全国·课后作业 查看更多[3]
北师大版九年级数学下册 第三章 圆 3.7 切线长定理 同步习题江苏省无锡市惠山区石塘湾中学2019-2020学年九年级上学期期中数学试题(已下线)辽宁省葫芦岛市连山区杨郊乡初级中学2021-2022学年九年级下学期第一次月考数学试题
更新时间:2018-03-05 14:48:18
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【推荐1】已知点P是线段AB上与点A不重合的一点,且AP<PB.AP绕点A逆时针旋转角α(0°<α≤90°)得到AP1,BP绕点B顺时针也旋转角α得到BP2,连接PP1、PP2.
![](https://img.xkw.com/dksih/QBM/2015/7/14/1573872109731840/1573872116097024/STEM/22f005b244ee40069a51cc3f3a473bbe.png?resizew=492)
(1)如图1,当α=90°时,求∠P1PP2的度数;
(2)如图2,当点P2在AP1的延长线上时,求证:△P2P1P∽△P2PA;
(3)如图3,过BP的中点E作l1⊥BP,过BP2的中点F作l2⊥BP2,l1与l2交于点Q,连接PQ,求证:P1P⊥PQ.
![](https://img.xkw.com/dksih/QBM/2015/7/14/1573872109731840/1573872116097024/STEM/22f005b244ee40069a51cc3f3a473bbe.png?resizew=492)
(1)如图1,当α=90°时,求∠P1PP2的度数;
(2)如图2,当点P2在AP1的延长线上时,求证:△P2P1P∽△P2PA;
(3)如图3,过BP的中点E作l1⊥BP,过BP2的中点F作l2⊥BP2,l1与l2交于点Q,连接PQ,求证:P1P⊥PQ.
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【推荐2】如图,
中,
,
,D,E分别是直线
,
边上的点,直线
,
交于点F.
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/6/e9286d9d-1b8b-466a-8f85-a9957a0cfd75.png?resizew=523)
(1)如图1,若
,求证:
;
(2)如图2,若
,求
的值;
(3)如图3,若
,
,求
的值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/047dc9795efa99b6fb9fdf9778085dab.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/597ce705d3a2fe04d29de9e81ec6250d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0dc5c9827dfd0be5a9c85962d6ccbfb1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2024/4/6/e9286d9d-1b8b-466a-8f85-a9957a0cfd75.png?resizew=523)
(1)如图1,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f0767d125e3f10a2a8ae97924ed94e55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b62dcef170114cd71b99f6646d364ef0.png)
(2)如图2,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1f56c68246102f861b8e10d1a18b0a39.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f4907f159ff9c4caacf266db591588.png)
(3)如图3,若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/eddbbc9ab5bfa1e9f5f583b44a506e52.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6d62be235d78fb8e8e05db0ac9d20700.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f4907f159ff9c4caacf266db591588.png)
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【推荐3】根据以下素材,探究完成任务
设计路的宽度 | ||
材料1 | 为培养学生劳动实践能力,某研学基地计划在一块形状为三角形的土地上开辟出一块矩形土地(如图所示)供种菜使用,其中![]() ![]() ![]() ![]() ![]() | |
材料2 | 为了方便学生使用,计划在开辟出来的长方形土地上建造三条如图所示的宽均为a(![]() | |
问题解决 | ||
任务1 | 若所开辟的土地为正方形,求该正方形![]() | |
任务2 | 若所开辟的土地为矩形,求矩形![]() | |
任务3 | 当![]() |
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【推荐1】在平面直角坐标系xOy中,对于任意三点A,B,C,给出如下定义:
如果矩形的任何一条边均与某条坐标轴平行,且A,B,C三点都在矩形的内部或边界上,则称该矩形为点A,B,C的覆盖矩形.点A,B,C的所有覆盖矩形中,面积最小的矩形称为点A,B,C的最优覆盖矩形.例如,下图中的矩形A1B1C1D1,A2B2C2D2,AB3C3D3都是点A,B,C的覆盖矩形,其中矩形AB3C3D3是点A,B,C的最优覆盖矩形.
(1)已知A(﹣2,3),B(5,0),C(t,﹣2).
①当t=2时,点A,B,C的最优覆盖矩形的面积为 ;
②若点A,B,C的最优覆盖矩形的面积为40,求直线AC的表达式;
(2)已知点D(1,1).E(m,n)是函数y=
(x>0)的图象上一点,⊙P是点O,D,E的一个面积最小的最优覆盖矩形的外接圆,求出⊙P的半径r的取值范围.
如果矩形的任何一条边均与某条坐标轴平行,且A,B,C三点都在矩形的内部或边界上,则称该矩形为点A,B,C的覆盖矩形.点A,B,C的所有覆盖矩形中,面积最小的矩形称为点A,B,C的最优覆盖矩形.例如,下图中的矩形A1B1C1D1,A2B2C2D2,AB3C3D3都是点A,B,C的覆盖矩形,其中矩形AB3C3D3是点A,B,C的最优覆盖矩形.
(1)已知A(﹣2,3),B(5,0),C(t,﹣2).
①当t=2时,点A,B,C的最优覆盖矩形的面积为 ;
②若点A,B,C的最优覆盖矩形的面积为40,求直线AC的表达式;
(2)已知点D(1,1).E(m,n)是函数y=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d1b7588653f727b6ff391077af2781e5.png)
![](https://img.xkw.com/dksih/QBM/2019/3/20/2164500290584576/2165530481737728/STEM/7eaa1e53411f4fef971e9919e3368b39.png?resizew=161)
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【推荐2】如图,在射线BA,BC,AD,CD围成的菱形ABCD中,∠ABC=60°,AB=6
,O是射线BD上一点,⊙O与BA,BC都相切,与BO的延长线交于点M.过M作EF⊥BD交线段BA(或射线AD)于点E,交线段BC(或射线CD)于点F.以EF为边作矩形EFGH,点G,H分别在围成菱形的另外两条射线上.
![](https://img.xkw.com/dksih/QBM/2016/8/16/1574230244655104/1574230250430464/STEM/e95e648b9aa84ad6b4a258e291eeb567.png?resizew=205)
(1)求证:BO=2OM.
(2)设EF>HE,当矩形EFGH的面积为24
时,求⊙O的半径.
(3)当HE或HG与⊙O相切时,求出所有满足条件的BO的长.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef812f839622326a7d7027cc806aaeb.png)
![](https://img.xkw.com/dksih/QBM/2016/8/16/1574230244655104/1574230250430464/STEM/e95e648b9aa84ad6b4a258e291eeb567.png?resizew=205)
(1)求证:BO=2OM.
(2)设EF>HE,当矩形EFGH的面积为24
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cef812f839622326a7d7027cc806aaeb.png)
(3)当HE或HG与⊙O相切时,求出所有满足条件的BO的长.
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【推荐3】【概念认识】
定义:对角线互相垂直且相等的四边形叫做垂等四边形.
(1)如图1,已知在垂等四边形
中,对角线
与
交于点
,若
,
cm,
,求
的长度,
【数学理解】
(2)在探究如何画“圆内接垂等四边形”的活动中,小李与同学讨论出了如下方法:如图2,在
中,已知
是
的弦,只需作
,
,分别交
于点
和点
,即可得到垂等四边形
,请你写出证明过程.
【问题解决】
(3)如图3,已知
是
上一定点,
为
上一动点,以
为一边作出
的内接垂等四边形(
、
不重合且
、
、
三点不共线),对角线
与
交于点
,
的半径为
,当点
到
的距离为
时,求弦
的长度.
定义:对角线互相垂直且相等的四边形叫做垂等四边形.
(1)如图1,已知在垂等四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1c26ed4600cef25f040caa553ac21051.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
【数学理解】
(2)在探究如何画“圆内接垂等四边形”的活动中,小李与同学讨论出了如下方法:如图2,在
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](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/719d071fa86799218653af756956b7aa.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/636379c98ab2f291a3c20ff1505b0416.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/c5db41a1f31d6baee7c69990811edb9f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
【问题解决】
(3)如图3,已知
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3d97cdc586744d208b6f69c9813af977.png)
![](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/5963abe8f421bd99a2aaa94831a951e9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f9e8449aad35c5d840a3395ea86df6d.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/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2a30f3a8b673cc28bd90c50cf1a35281.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/03902478df1a55bc99703210bccab910.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a7ffe8515ff6183c1c7775dc6f94bdb8.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/22/204007ab-bd96-4566-99d5-827c14bef584.png?resizew=386)
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