Question

（本题满分7分）将一条长为20cm的铁丝剪成两段，并以每一段铁丝的长度为周长做成一个正方形．
（1）要使这两个正方形的面积之和等于，那么这段铁丝剪成两段后的长度分别是多少？
（2）两个正方形的面积之和可能等于吗？若能，求出两段铁丝的长度；若不能，请说明理由．

Answer 1

解：设这段铁丝被分成两段后，围成正方形．其中一个正方形的边长为，则另一个正方形的边长为．············································································· 1分
依题意列方程得：
，················································································· 3分
解方程得：，
因此这段铁丝剪成两段后的长度分别是4cm、16cm．·········································· 5分
（2）两个正方形的面积之和不可能等于．理由：
设两个正方形的面积和为，则：
，····························································· 7分
，当，的最小值．
两个正方形的面积之和不可能等于．·················································· 9分
（另解：由（1）可知：，···················································· 6分
化简后得：，
，
方程无实数解，
所以两个正方形的面积之和不可能等于．）·············································· 9分

解析