类比、转化、从特殊到一般等思想方法,在数学学习和研究中经常用到,如下是一个案例,请补充完整.
原题:如图1,在▱ABCD中,点E是BC边上的中点,点F是线段AE上一点,BF的延长线交射线CD于点G,若
=3,求
的值.
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/3247161cded5439797fb299dd74611b1.png)
(1)尝试探究
在图1中,过点E作EH∥AB交BG于点H,则AB和EH的数量关系是 ,CG和EH的数量关系是 ,
的值是
(2)类比延伸
如图2,在原题的条件下,若
=m(m≠0),则
的值是 (用含m的代数式表示),试写出解答过程.
(3)拓展迁移
如图3,梯形ABCD中,DC∥AB,点E是BC延长线上一点,AE和BD相交于点F,若
=a,
=b(a>0,b>0),则
的值是 (用含a,b的代数式表示).
原题:如图1,在▱ABCD中,点E是BC边上的中点,点F是线段AE上一点,BF的延长线交射线CD于点G,若
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/3314524935a44c79b827ccb82596fab0.png)
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/2c3878edc864473f98c20fb9c835a298.png)
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/3247161cded5439797fb299dd74611b1.png)
(1)尝试探究
在图1中,过点E作EH∥AB交BG于点H,则AB和EH的数量关系是 ,CG和EH的数量关系是 ,
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/2c3878edc864473f98c20fb9c835a298.png)
(2)类比延伸
如图2,在原题的条件下,若
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/3314524935a44c79b827ccb82596fab0.png)
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/1da6e0c0889a4789b6a14f2623723042.png)
(3)拓展迁移
如图3,梯形ABCD中,DC∥AB,点E是BC延长线上一点,AE和BD相交于点F,若
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/69b5722eb02f4009bd8f8accf01f0065.png)
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/f8325de18dda4c188aa6fc30ae5ac766.png)
![](https://img.xkw.com/dksih/QBM/2016/4/19/1574194810609664/1574194816245760/STEM/fdbafd4957394e08853341cc51b4bb33.png)
更新时间:2016-12-06 12:10:27
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【知识点】 相似三角形的判定与性质综合
相似题推荐
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【推荐1】【问题发现】
![](https://img.xkw.com/dksih/QBM/2021/9/9/2804289867153408/2804297435684864/STEM/c7af429d-652b-433e-ad25-d5d34d625e7e.png?resizew=691)
(1)若四边形
是菱形,
,点P是射线
上一动点,以
为边向右侧作等边
,如图1,当点E在菱形
内部或边上时,连接
,则
与
有怎样的数量关系?并说明理由;
【类比探究】
(2)若四边形
是正方形,点P是射线
上一动点,以
为直角边在
边的右侧作等腰
,其中
,如图2.当点P在对角线
上,点E恰好在
边所在直线上时,则
与
之间的数量关系?并说明理由;
【拓展延伸】
(3)在(2)的条件下,如图3,在正方形
中,
,当P是对角线
的延长线上一动点时,连接
,若
,求
的面积.
![](https://img.xkw.com/dksih/QBM/2021/9/9/2804289867153408/2804297435684864/STEM/c7af429d-652b-433e-ad25-d5d34d625e7e.png?resizew=691)
(1)若四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e075468e7fb0bf30229aec01a7205977.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/20a541b81584a032f571159ea152c85a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/729684f958ad60b8e905fe1e1da53c03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b3ce322b763173805d32db4677ef8094.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
【类比探究】
(2)若四边形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](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/5774d8bb7437c44f915cfb77b968960a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/00261243d533be20524a309568e0a5b9.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/9d78abbad68bbbf12af10cd40ef4c353.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2cdba1337ec85fa9722cb4b320a82ae6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4eedae8d316c76e3d0b451256de03fb9.png)
【拓展延伸】
(3)在(2)的条件下,如图3,在正方形
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/411b38a18046fea8e9fab1f9f9b80a5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/080ca48cd27d4bf9d9ef084b558fc17a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d40b319212a7e7528b053e1c7097e966.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/85c4bdfb0db1e31e8459df1d15f9ab55.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/37b7bd8a2436e2f2bb89fefa70139273.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ad3cd61d00f89e68ccca2cac5c937783.png)
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【推荐2】如图,在直角坐标系中,
,点B绕着点A顺时针旋转
得到![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/15/9600cd66-d44e-49f1-bf14-f08b20bb3f6d.png?resizew=339)
(1)求直线
的表达式
(2)求直线
的表达式
(3)若平面内有一点C,使
是以
为腰的等腰三角形且与
相似,求点C的坐标
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f402b422215aa4121999c3e4fd562537.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f6b86c22b670a8e9f3896f9e8883fbbb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3953cec61ac602ce5eb59b7912352179.png)
![](https://img.xkw.com/dksih/QBM/editorImg/2022/11/15/9600cd66-d44e-49f1-bf14-f08b20bb3f6d.png?resizew=339)
(1)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
(2)求直线
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/943712a5e96b16cc15d775cc4687237e.png)
(3)若平面内有一点C,使
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/f52a58fbaf4fea03567e88a9f0f6e37e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/19343455668abab3ca3b05aa2cf616c2.png)
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【推荐3】已知抛物线对应的二次函数为y=a(x+10)(x+5),它与x轴交于A、B两点(A在B的左侧),与y轴交于C点,点D是以B为圆心、5为半径的圆周上位于第二象限内的动点,直线AD与y轴交于点E,设E(0,2t).
![](https://img.xkw.com/dksih/QBM/2022/4/15/2958963717406720/2961289881673728/STEM/860e3047-4abd-4207-ba7a-b124cbab0589.png?resizew=224)
(1)在抛物线对称轴上分别求满足下列条件的点的坐标(用t表示):
①求点P使△PBE的周长最小;
②求点Q使QE-QB的值最大;
(2)若直线CD与⊙B相切,试用t表示a;
(3)在(1)、(2)的条件下,若6≤OD≤8,求△CPB面积的取值范围.
![](https://img.xkw.com/dksih/QBM/2022/4/15/2958963717406720/2961289881673728/STEM/860e3047-4abd-4207-ba7a-b124cbab0589.png?resizew=224)
(1)在抛物线对称轴上分别求满足下列条件的点的坐标(用t表示):
①求点P使△PBE的周长最小;
②求点Q使QE-QB的值最大;
(2)若直线CD与⊙B相切,试用t表示a;
(3)在(1)、(2)的条件下,若6≤OD≤8,求△CPB面积的取值范围.
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