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). （2分）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．
3在（2）的條件下，若為上一點且，拋物線經(jīng)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(biāo)；若不存在，請說明理由．

Answer 1

p;【答案】

(1). （2分）試比較、的大小，并說明理由．
（1），理由如下：
由折疊知：  在中，為斜邊 
故················································································································· 2分
(2). （1分）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分
(3).在（2）的條件下，若為上一點且，拋物線經(jīng)過、兩點，請求出此拋物線的解析式．
，，
       

為等邊三角形，················································································ 4分
作于．     
 的坐標(biāo)為·································································· 5分
拋物線過點，，
   
所求拋物線解析式為········································································ 6分
(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(biāo)；若不存在，請說明理由．
由（3）：
當(dāng)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(biāo)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(dāng)時，
當(dāng) ，
，…………………10分解析:
略