Question

以▱ABCD的四条边为边，在其形外分别作正方形，如图，连接EF、GH、IJ、KL．若▱ABCD的面积为5，则图中阴影部分四个三角形的面积和为（　　）                                                              

                                                                         

A．5                            B．10                           C．15                          D．20

Answer 1

B【考点】平行四边形的性质．                                                                

【分析】过D作DN⊥AB于N，过E作EM⊥FA交FA延长线于M，连接AC，BD，求出∠EAM=∠BAD，根据锐角三角形函数定义求出EM=DN，求出△AEF和△ABD面积相等，同理求出S_△BHG=S_△ABC，S_△CIJ=S_△CBD，S_△DLK=S_△DAC，代入S=S_△AEF+S_△BGH+S_△CIJ+S_△DLK得出S=2S_{平行四边形ABCD}，代入求出即可．                

【解答】解：过D作DN⊥AB于N，过E作EM⊥FA交FA延长线于M，连接AC，BD，                  

∵四边形ABGF和四边形ADLE是正方形，                                             

∴AE=AD，AF=AB，∠FAB=∠EAD=90°，                                               

∴∠EAF+∠BAD=360°﹣90°﹣90°=180°，                                                

∵∠EAF+∠EAM=180°，                                                                         

∴∠EAM=∠DAN，                                                                           

∴sin∠EAM=，sin∠DAN=，                                                          

∵AE=AD，                                                                                        

∴EM=DN，                                                                                       

∵S_△AEF=AF×EM，S_△ADB=AB×DN，                                                  

∴S_△AEF=S_△ABD，                                                                              

同理S_△BHG=S_△ABC，S_△CIJ=S_△CBD，S_△DLK=S_△DAC，                                     

∴阴影部分的面积S=S_△AEF+S_△BGH+S_△CIJ+S_△DLK=2S_{平行四边形ABCD}=2×5=10．                

故选B．                                                                                            

                                                                         

【点评】本题考查了平行四边形的性质，锐角三角函数的定义，三角形的面积等知识点的应用，主要考查学生运用定理进行推理和计算的能力，题目比较好，但有一定的难度．