Question

如圖所示，將矩形沿折疊，使點恰好落在上處，以為邊作正方形，延長至，使，再以、為邊作矩形．

(1).試比較、的大小，并說明理由．

(2).令，請問是否為定值？若是，請求出的值；若不是，請說明理由．

為定值．

(3).在（2）的條件下，若為上一點且，拋物線經過、兩點，請求出此拋物線的解析式．

(4).在（3）的條件下，若拋物線與線段交于點，試問在直線上是否存在點，使得以、、為頂點的三角形與相似？若存在，請求直線與軸的交點的坐標；若不存在，請說明理由．

 

 

 

 

 

 

 

 

 

 

 

Answer 1

解：（1），理由如下：

由折疊知：   在中，為斜邊 

故····························

（2）

···································································································· 3分

（3），，

        

為等邊三角形，················································································ 4分

作于．      

  的坐標為·································································· 5分

拋物線過點，，

   

所求拋物線解析式為········································································ 6分

（4）由（3）：

當時，

·························································· 7分

方法1：若與相似，

而．則分情況如下

時為或····························· 8分

時    為或（0，1）······································ 9分

故直線與軸交點的坐標為或或或（0，1）···············10分

方法2：與相似時，由（3）得則或，

過點作垂直軸于則或

當時，

當 ，

， …………………10分

 

解析:略