题目
题型:不详难度:来源:
活动情境:
如图2,将边长为8cm的正方形纸片ABCD沿EG折叠(折痕EG分别与AB、DC交于点E、G),使点B落在AD边上的点 F处,FN与DC交于点M处,连接BF与EG交于点P.
所得结论:
当点F与AD的中点重合时:(如图1)甲、乙、丙三位同学各得到如下一个正确结论(或结果):
甲:△AEF的边AE= cm,EF= cm;
乙:△FDM的周长为16 cm;
丙:EG=BF.
你的任务:
小题1:填充甲同学所得结果中的数据;
小题2: 写出在乙同学所得结果的求解过程;
小题3:当点F在AD边上除点A、D外的任何一处(如图2)时:
① 试问乙同学的结果是否发生变化?请证明你的结论;
② 丙同学的结论还成立吗?若不成立,请说明理由,若你认为成立,先证明EG=BF,再求出S(S为四边形AEGD的面积)与x(AF=x)的函数关系式,并问当x为何值时,S最大?最大值是多少?
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000602-37121.png)
答案
小题1:AE= 3 cm, EF= 5 cm;设AE=x,则EF=8-x,AE=4,∠A=90°,
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000603-85707.png)
小题2:解:如答图1,∵∠MFE=90°,∴∠DFM+∠AFE=90°,
又∵∠A=∠D=90°,∠AFE=∠DMF,∴△AEF∽△DFM,∴
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000603-17549.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000603-98716.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000603-49768.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000604-29950.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000604-88327.png)
∴△FMD的周长=4+
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000604-19475.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000604-49850.png)
小题3:① 乙的结果不会发生变化
理由:如答图2,设AF=x,EF=8-AE,
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-84287.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-20273.png)
同上述方法可得△AEF∽△DFM,
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-87568.png)
则
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-84757.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000606-77579.png)
② 丙同学的结论还成立
证明:如答图2,∵B、F关于GE对称,∴BF⊥EG于P,过G作GK⊥AB于K,∴∠FBE=∠KGE,
在正方形ABCD中,GK=BC=AB,∠A=∠EKG=90°,∴△AFB≌△KEG,∴FB=GK.由上述可知AE=4-
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-20273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-20273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000606-73223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-20273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000605-20273.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000606-35358.png)
S =
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000606-71320.png)
当x=4,即F与AD的中点重合时,
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000607-61355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000607-61355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000607-12952.png)
解析
核心考点
试题【某班甲、乙、丙三位同学进行了一次用正方形纸片折叠探究相关数学问题的课题学习活动.活动情境:如图2,将边长为8cm的正方形纸片ABCD沿EG折叠(折痕EG分别与A】;主要考察你对相似图形等知识点的理解。[详细]
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000558-47742.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000558-28543.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000558-93966.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000558-52194.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000559-88263.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000559-35562.png)
A.2 | B.4 |
C.6 | D.8 |
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000550-60893.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000551-88322.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000547-80637.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000547-70700.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000547-49995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000548-71481.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000528-81080.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000529-32868.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000529-37875.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000529-75241.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000529-98186.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-79835.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-66995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-80677.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-61808.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-66995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000531-22064.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000531-29140.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000531-35607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000531-62261.jpg)
小题1:(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000532-17838.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000532-59521.png)
小题2:(2)记
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000532-53901.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000532-53155.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000533-75195.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000533-10270.png)
小题3:(3)请探索:是否存在这样的点
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-66995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000533-36780.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000533-91375.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000534-56798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000534-34598.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000530-66995.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000524-82345.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191103/20191103000524-35513.png)
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