Question

如圖所示，將矩形沿折疊，使點(diǎn)恰好落在上處，以為邊作正方形，延長(zhǎng)至，使，再以、為邊作矩形．

(1).試比較、的大小，并說明理由．
(2).令，請(qǐng)問是否為定值？若是，請(qǐng)求出的值；若不是，請(qǐng)說明理由．
為定值．
(3).在（2）的條件下，若為上一點(diǎn)且，拋物線經(jīng)過、兩點(diǎn)，請(qǐng)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(diǎn)，試問在直線上是否存在點(diǎn)，使得以、、為頂點(diǎn)的三角形與相似？若存在，請(qǐng)求直線與軸的交點(diǎn)的坐標(biāo)；若不存在，請(qǐng)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(biāo)為·································································· 5分
拋物線過點(diǎn)，，
   
所求拋物線解析式為········································································ 6分
（4）由（3）：
當(dāng)時(shí)，
·························································· 7分

方法1：若與相似，
而．則分情況如下
時(shí)為或····························· 8分
時(shí)   為或（0，1）······································ 9分
故直線與軸交點(diǎn)的坐標(biāo)為或或或（0，1）··············· 10分
方法2：與相似時(shí)，由（3）得則或，
過點(diǎn)作垂直軸于則或
當(dāng)時(shí)，
當(dāng) ，
，…………………10分解析:
略