题目
如图所示,将矩形沿折叠,使点恰好落在上处,以为边作正方形,延长至,使,再以、为边作矩形.(1).试比较、的大小,并说明理由.(2).令,请问是否为定值?若是,请求出的值;若不是,请说明理由.为定值.(3).在(2)的条件下,若为上一点且,抛物线经过、两点,请求出此抛物线的解析式.(4).在(3)的条件下,若抛物线与线段交于点,试问在直线上是否存在点,使得以、、为顶点的三角形与相似?若存在,请求直线与轴的交点的坐标;若不存在,请说明理由.
答案:解:(1),理由如下:由折叠知: 在中,为斜边 故····························(2)···································································································· 3分(3),, 为等边三角形,················································································ 4分作于. 的坐标为·································································· 5分抛物线过点,, 所求抛物线解析式为········································································ 6分(4)由(3):当时,·························································· 7分方法1:若与相似,而.则分情况如下时为或····························· 8分时 为或(0,1)······································ 9分故直线与轴交点的坐标为或或或(0,1)···············10分方法2:与相似时,由(3)得则或,过点作垂直轴于则或当时,当 ,, …………………10分 解析:略