Question

如图,将长方形纸片ABCD沿BD折叠,使点C落在C'处,BC‘交AD于E,若AD=8,AB=4,求S△BED

Answer 1

将长方形纸片ABCD沿BD折叠,使点C落在C'处,BC‘交AD于E,若AD=8,AB=4,求S△BED
解析：由题意设∠BDC=a,∠ADB=b
∵ABCD为矩形,∴a+b=90°
∵沿BD折叠,使点C落在C'处,BC‘交AD于E
∴⊿BDC’≌ ⊿BDC
∠BDC=∠BDC’=a
∴∠C’DA=∠BDC’-∠BDA=a-b=90°-2b
∵AD=8,AB=4,
∴BD=√(8^2+4^2)=4√5,sinb=AB/BD=4/(4√5)= √5/5,cosb=AD/BD=8/(4√5)=2√5/5
Cos(a-b)=cos(90°-2b)=DC’/DE==>DE=4/ cos(90°-2b)=4/sin(2b)
Sin(2b)=2sinbcosb=4/5==>DE=5
∴S(⊿BED)=1/2*DE*AB=1/2*5*4=10