1.AB^2=AC^2+BC^2
BE^2=BC^2+EC^2, 16=BC^2+AC^2/4 .(1)
AD^2=AC^2+CD^2, 49=AC^2+BC^2/4 .(2)
(1)+(2): 65=5/4AC^2+5/4BC^2, AC^2+BC^2=65*4/5=52
AB^2=52, AB=√52=2√13
2.DE⊥DF, 四边形EADF为长方形
D为AB的中点
所以E为AC的中点, F为BC的中点
所以EF∥BC, DE∥BC, DF∥AC
DF=AE, DE=BF
EF^2=DE^2+DF^2=AE^2+BF^2
3.△ABC为等腰直角三角形
过A作BC的垂线,垂足为M, 则AM=MC
AD^2=AM^2+DM^2=MC^2+DM^2 .(1)
CD^2=(DM+MC)^2
BD^2=(BM-DM)^2=(MC-DM)^2
CD^2+BD^2
=(DM+MC)^2+(MC-DM)^2
=DM^2+2DM*MC+MC^2+MC^2-2MC*DM+DM^2
=2(DM^2+MC^2)
=2AD^2 (将(1)代入)
4.△ABE中,角A=60度,角AEB=90度,则角ABE=30度
AB=4, AE=AB/2=2, BE=ABsin60度=√3/2*4=2√3
△BCE中, BE=BC=2√3, 角EBC=60度
所以△BCE为等边三角形
CE=BC=2√3
△CDE中, DE=3, CE=2√3, CD=√3
DE^2+CD^2=3^2+(√3)^2=9+3=12=(2√3)^2=CE^2
CD⊥DE, 即AD⊥CD
5.