Question

如图,在梯形ABCD中,AD∥BC,AB=DC=AD,∠C=60°,AE⊥BD于点E,AE=1.求梯形ABCD的面积.

Answer 1

过D作DF∥AB交BC于F,

∵AD∥BC,∴四边形ABFD是平行四边形,∴DF=AB,又AB=CD,

∴DF=CD,∵∠C=60°,∴ΔDFC是等边三角形,

∴∠ABC=60°,

∵AB=AD,∴ABFD是菱形,∠ADB=∠ABD,又∠ADB=∠CBD,

∴∠ABD=∠CBD=1/2∠ABC=30°,

∴AB=2AE=2,

∴SΔDFC=√3/4×2^2=√3,

又S菱形ABFD=2SΔDFC=2√3,

∴S梯形=S菱形ABFD+SΔDFC=3√3.