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1、初中几何的最值问题，主要是求一条或两条线段长度的最大（最小）值，三角形或四边形周长的最小值，对一些简单问题可以通过诸如“两点之间线段最短”“垂线段最短”等定理解决，综合近几年中考常见的同类考题，经常用到的解决方法主要有以下4 种：1、垂线段最短 2、利用轴对称3、构造三角形，巧用三角形三边关系4、巧用辅助圆 5、构造函数关系。对每类问题的解决方法及规律，通过以下例题说明。一、利用垂线段最短 解决问题1、如图，边长为 6 的等边三角形 ABC 中，E 是对称轴 AD 上的一个动点，连接 EC，将线段 EC 绕点 C 逆时针旋转 60得到 FC，连接 DF 则在点 E 运动过程中，DF 的最小值是
2、二、轴对称PQACDABCBPPPBABPABPQAB1、（八年级上册数学课本90 页第 18 题）如图,甲、乙两个单位分别位于一条封闭式街道的两旁,现准备合作修建一座过街天桥.问：（1）桥建在何处才能使由甲到乙的路线最短?（注：桥必须与街道垂直）（2）桥建在何处才能使甲、乙到桥的距离相等?三、构造三角形，巧用三角形三边关系文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP1
3、0U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP
4、10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:C
5、P10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:
6、CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码
7、:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编
8、码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档
9、编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F21、如图，正方形 ABCD 中，AB=8，O 为 AB 的中点，P 为正方形 ABCD 外一动点，且 APCP，则线段 OP 的最大值为（）A.4+4 B.2 C.4 D.6 第 1 题第 2 题2、如图，E，F 是正方形ABCD 的边 AD 上两个动点，满足AE=DF 连接 CF 交 BD 于 G，连接 BE 交 AG 于点 H若正方形的边长
10、为 2，则线段 DH 长度的最小值是 _ 3、如图，E、F 分别是边长为 2 的正方形 ABCD 边 AD、AB 上的两个动点，满足AE+AF=2，BE 交 CF 于点 P，在点 E、F 运动过程中，PA的最小值是多少？四、巧用辅助圆1、如图，ABC中，ABC=90，AB=6，BC=8，O 为 AC 的中点，过D 作 OE OF，OE、OF 分别交射线AB、BC 于 E、F，则 EF 的最小值为 _ 文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T
11、7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7
12、T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U
13、7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10
14、U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP1
15、0U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP
16、10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:C
17、P10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2五、构造函数关系1、如图，正方形ABCD 的边长为4，E、F 分别是 BC、CD 上的两个动点，且AEEF则AF 的最小值是 _.答案一、利用垂线段最短 解决问题解答：法1：连接 BF 文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5
18、F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q
19、5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2
20、Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z
21、2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10
22、Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G1
23、0Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G
24、10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2法 2：如图，取AC 的中点 G，连接 EG，旋转角为60，ECD+DCF=60 ，又 ECD+GCE=ACB=60 ，DCF=GCE，AD 是等边 AB
25、C 的对称轴，CD=12BC，CD=CG，又 CE 旋转到 CF，CE=CF，在DCF 和 GCE 中，CECF DCF GCECD CG，DCF GCE（SAS），DF=EG，根据垂线段最短，EGAD 时，EG 最短，即DF 最短，此时 CAD=12 60=30，AG=12AC=126=3，EG=12AG=123=1.5，DF=1.5 二、轴对称三、构造三角形，巧用三角形三边关系1、文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA
26、5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 H
27、A5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7
28、HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7
29、 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M
30、7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9
31、M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A
32、9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2连接 AC、BD 相交于 Q，连接 PQ，ABCD是正方形，AQ=CQ，AP CP，PQ=1/2AC=42，O为 AB 中点，OQ=1/2BC=4，OP OQ+PQ=4+4 2，当 OP过 Q 时，OP最大=4+42。2、文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 Z
33、A10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5
34、ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5
35、 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F
36、5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3
37、F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z
38、3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3
39、Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F23、取 BC的中点 M，连接 AM、PM，构造三角形APM 四、巧用辅助圆1、解法两种是过点 O 作 OM AB 于点 M，作 ON
40、BC 于点 N，ABC=90，四边形 OMBN是矩形，OM BC，ON AB，AOM ACB，CON CAB，OM：BC=OA：AC，ON：AB=OC：AC，O 为 AC 的中点，OM=3，MN=5，由垂线段最短，可得当OE 与 OM重合，即EF 与 MN重合时，EF 最短，EF 的最小值为5是作辅助圆五、构造函数关系1、设 BE=x，则 EC=4 x，先利用等角的余角相等得到BAE=FEC，则可判断Rt ABERt ECF，利用相似比可表示出FC=，则DF=4 FC=4=x2x+4=（x2）2+3，所以 x=2 时，DF 有最小值3，而 AF2=AD2+DF2，即 DF 最小时，AF 最小，
41、AF 的最小值为=5 解：设 BE=x，则 EC=4 x，AEEF，AEF=90，AEB+FEC=90，而AEB+BAE=90，BAE=FEC，文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7
42、A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T
43、7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7
44、T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U
45、7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10
46、U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP1
47、0U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP
48、10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2 Rt ABE Rt ECF，=，即=，解得 FC=，DF=4 FC=4=x2x+4=（x2）2+3 当 x=2 时，DF 有最小值3，AF2=AD2+DF2，AF 的最小值为=5 文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文
49、档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2
50、文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F2文档编码:CP10U7T7A9M7 HA5M4S3Z3F5 ZA10G10Z2Q5F