(1)证明:连结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149259531.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230141492903960.png)
由题意得,------------1分
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149306621.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149321883.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149337373.png)
为公共边
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149415712.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149446843.png)
-------------------2分
(利用勾股定理逆定理相应给分)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149462537.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
与圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149493292.png)
相切.-------------------3分
(2)当点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149524318.png)
运动到与
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149540309.png)
点重合的位置时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230141495556156.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
为正方形
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149696534.png)
的对角线,所以此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
最长,有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149727955.png)
-----------------4分
当点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149524318.png)
运动到线段
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149337373.png)
与半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149493292.png)
的交点处时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
最短.
-----------------5分
证明如下:
在半圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149493292.png)
上任取一个不与点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149524318.png)
重合的点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149836335.png)
,连结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149852409.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149867423.png)
.
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149883561.png)
中,∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149898676.png)
即:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149914684.png)
,
∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149930468.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149961508.png)
∵点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149836335.png)
是任意一个不与点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149524318.png)
重合的点,∴此时
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
最短. -----------------6分
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230141500541654.png)
-------------7分
(3)当点E与点A重合时,DE=DA=10,此时,直线DE的解析式为
y=10;
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230141500704402.png)
---------8分
当点
E与点
A不重合时,过点
E作
GH⊥
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150086266.png)
轴,分别交
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150210389.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150086266.png)
轴于点
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150304316.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150320303.png)
,连结
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150335399.png)
.
则四边形
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150351551.png)
是矩形,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
为圆
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149493292.png)
的切线
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150600736.png)
=90°
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150616736.png)
-----------------------9分
又∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150632898.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150647537.png)
∽
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150663561.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150678947.png)
----------------------10分
设
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150694614.png)
,则有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150694514.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150710730.png)
得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150725837.png)
,-----------------------11分
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150741728.png)
, 即:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150756566.png)
----------------12分
又直线
DE过点
D(10,10),设直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
解析式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150788585.png)
,则有:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150803888.png)
,
解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150819954.png)
,即:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149228715.png)
∴当
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150850506.png)
时,直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149477408.png)
的解析式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149228715.png)
或
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014149243422.png)
-----------------------14分
以下两种解法涉及高中知识,仅供参考:
另解2:
(1)当点E与点A重合时,DE=DA=10,此时,直线DE的解析式为
y=10;
(2)当点E与点A不重合时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150912820.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/201408230141509281845.png)
设直线
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150944678.png)
且经过点(10,10),代入求得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150959532.png)
所以直线DE的解析式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150975716.png)
另解3:
依题意得:点
O的坐标为(0,5),设直线
DE的解析式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150975575.png)
由点到直线的距离公式得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014150990768.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014151006815.png)
①
直线
DE过点
D(10,10),得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014151022574.png)
②
由①②解得:
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014151037790.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014151053617.png)
所以直线
DE的解析式为
![](http://thumb.1010pic.com/pic2/upload/papers/20140823/20140823014151068847.png)
(1)如图1,连接OE,OD,由题意得,DE=DA=10,利用(SSS)判定△AOD≌△EOD,从可得∠OED=∠OAD=90°即可.
(2)当点E运动到与B点重合的位置时,如图2,DE为正方形ABCD的对角线,所以此时DE最长,利用勾股定理求得DE,证明当点E运动到线段OD与半圆O的交点处时,DE最短.然后求得DE=OD-OE即可.
(3)当点E与点A重合时,DE=DA=10,此时,直线DE的解析式为y=10;如图4,当点E与点A不重合时,过点E作GH⊥x轴,分别交AD,x轴于点G,H,连接OE.则四边形AFEG是矩形,且DE为圆O的切线,求证△OFE∽△DGE,利用其对应边成比例,设E(m,n),则有:EF=m,OF=OB-FB=5-n求得即可