问题背景:
如图①,在四边形ADBC中,∠ACB=∠ADB=90°,AD=BD,探究线段AC,BC,CD之间的数量关系.
小吴同学探究此问题的思路是:将△BCD绕点D,逆时针旋转90°到△AED处,点B,C分别落在点A,E处(如图②),易证点C,A,E在同一条直线上,并且△CDE是等腰直角三角形,所以CE=
CD,从而得出结论:AC+BC=
CD.
简单应用:
(1)在图①中,若AC=
,BC=
,则CD= .
(2)如图③,AB是⊙O的直径,点C、D在⊙上,
,若AB=13,BC=12,求CD的长.
拓展规律:
(3)如图④,∠ACB=∠ADB=90°,AD=BD,若AC=m,BC=n(m<n),求CD的长(用含m,n的代数式表示)
(4)如图⑤,∠ACB=90°,AC=BC,点P为AB的中点,若点E满足AE=
AC,CE=CA,点Q为AE的中点,则线段PQ与AC的数量关系是 .
如图①,在四边形ADBC中,∠ACB=∠ADB=90°,AD=BD,探究线段AC,BC,CD之间的数量关系.
小吴同学探究此问题的思路是:将△BCD绕点D,逆时针旋转90°到△AED处,点B,C分别落在点A,E处(如图②),易证点C,A,E在同一条直线上,并且△CDE是等腰直角三角形,所以CE=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
简单应用:
(1)在图①中,若AC=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf298f00799cbf34b4db26f5f63af92f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/95bacae35b6e16a0a33c2bdc6bc07df7.png)
(2)如图③,AB是⊙O的直径,点C、D在⊙上,
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33ed7241864914faa9b2e12314e65c3c.png)
拓展规律:
(3)如图④,∠ACB=∠ADB=90°,AD=BD,若AC=m,BC=n(m<n),求CD的长(用含m,n的代数式表示)
(4)如图⑤,∠ACB=90°,AC=BC,点P为AB的中点,若点E满足AE=
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4dac452fbb5ef6dd653e7fbbef639484.png)
![](https://img.xkw.com/dksih/QBM/2016/8/3/1574223025315840/1574223031984128/STEM/1d80a6fd398d40ff9782967be8bfc984.png)
2016·江苏淮安·中考真题 查看更多[6]
2016年初中毕业升学考试(江苏淮安卷)数学(已下线)决胜2018中考压轴题全揭秘 专题17 圆问题(已下线)决胜2018中考压轴题全揭秘 专题28 探究型问题专题四阅读理解类例题(已下线)3.4圆周角和圆心角的关系(重点练)-2020-2021学年九年级数学下册十分钟同步课堂专练(北师大版)江苏省苏州市工业园区星汇中学2020-2021学年九年级下学期3月月考数学试卷
更新时间:2016-12-06 13:07:52
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【推荐1】如图,AB是半圆O的直径,C是
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解题方法
【推荐2】如图,四边形ABCD内接于⊙O,对角线AC是⊙O的直径,过点C作AC的垂线交AD的延长线于点E,F为CE的中点,连接BD,DF,BD与AC交于点P.
![](https://img.xkw.com/dksih/QBM/2022/2/24/2923424571121664/2927595136073728/STEM/4bf082edef6747949c25a39d69a0cafb.png?resizew=144)
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DE,求tan∠ABD的值;
(3)若∠DPC=45°,PD2+PB2=8,求AC的长.
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(1)求证:DF是⊙O的切线;
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【推荐1】在
中,
,
,在平面内,把
绕点
旋转得到
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垂直直线
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的延长线交
于点
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![](https://img.xkw.com/dksih/QBM/editorImg/2023/11/27/74502bfb-9f85-4c96-90c9-902d94ccb510.png?resizew=333)
(1)如图①,若
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是等腰三角形;
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(1)如图①,若
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e7d480e8b2c516612b47a19cb62d9bb0.png)
(2)如图②,若点
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(3)连接
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
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【推荐2】综合与实践
问题情境:
在“综合与实践”活动课上,老师给出了一张如图1所示的正方形纸片
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在线段
上,点
在线段
上,且满足
,连接
.
数学思考:
与
的数量关系为___________,位置关系为___________.
猜想证明:
(2)如图2,连接
交
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拓展探究:
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绕点
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![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49b50357a6545cae8348e3059312f520.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/60ef95894ceebaf236170e8832dcf7e3.png)
猜想证明:
(2)如图2,连接
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【推荐3】点A、B、C在同一直线上,在直线AC的同侧作△ABE和△BCF,连接AF,CE.取AF、CE的中点M、N,连接BM,BN,MN.
(1)若△ABE和△FBC是等腰直角三角形,且∠ABE=∠FBC=90°(图1),则△MBN是______三角形;
(2)在△ABE和△BCF中,若BA=BE,BC=BF,且∠ABE=∠FBC=α,(图2),则△MBN是______三角形,且∠MBN=______;
(3)若将(2)中的△ABE绕点B旋转一定角度,(图3),其他条件不变,那么(2)中的结论是否成立?若成立,给出你的证明;若不成立,写出正确的结论并给出证明.
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