如圖所示，直線l過正方形ABCD的頂點B，點A、C到直線l的距離分別為1和3，則正方形ABCD的邊長是（　　）...

問題詳情：

如圖所示，直線l過正方形ABCD的頂點B，點A、C到直線l的距離分別為1和3，則正方形ABCD的邊長是（　　）                                                                        

A．2                       B．3                            C．                       D．4

【回答】

C【分析】設BE=x，BF=y，先*Rt△BEA∽Rt△CFB，由相似的*質得xy=3…①，再由勾股定理得1+x2=32+y2…②，聯立①②解方程組即可．                                                                       

【解答】解：設BE=x，BF=y，                                                               

∵易*Rt△BEA∽Rt△CFB，                                                                   

∴，                                                                                      

∴xy=3…①                                                                                        

∵正方形ABCD中：AB=BC                                                                     

∴1+x2=32+y2…②                                                                               

由①可知x=，將其代入化簡得：y4+8y2﹣9=0                                              

解之、檢驗符合題意的：y=1，                                                                

∴x=3，y=1                                                                                        

AC2=1+x2=10，                                                                                  

∴AC=                                                                                         

即：正方形的邊長為：                                                                     

故：選C                                                                                            

【點評】本題考查了正方形的*質與相似三角形的判定與*質，解題的關鍵是分析圖形中存在的等量關係及有數形結合的思想意識．                                                                                        

知識點：相似三角形

題型：選擇題