Question

已知,如图,在△ABC中,∠ABC=90°,AB=AC,点D,E分别在AC,AB上,且DE⊥AC,联接CE,取CE的中点

Answer 1

∵DE⊥AC,∠ABC=90°

M是CE的中点

∴在RT△CDE和RT△CBE中

DM=1/2CE=CM

BM=1/2CE=CM

∴DM=BM

∵BM=CM

CM=DM

∴∠MDC=∠MCD

∠MBC=∠MCB

∵AB=BC

∴∠ACB=∠CAB=45°

∵∠ACB=∠MCD+∠MCB=45°

∴∠MDC+∠MBC=45°

∴∠AMC+∠DMC=180°-∠MDC-∠MCD+180°-∠MBC-∠MCB

=360°-(∠MCD+∠MCB)-(∠MDC+∠MBC)

=360°-45°-45°

=270°

∴∠BMD=360°-(∠AMC+∠DMC)=360°-270°=90°

∴BM⊥DM