Mathematics 18 of Harbin senior high school entrance examination in 2009.
Because there is BF=AE, you can set AF=3, so there is a rectangular ABEF. To find BM is to find the half-length of the diagonal of the rectangle ABE, and the triangle Abe is a right triangle. According to Pythagoras' law, AE length is 5, AE=BF, BF=2BM, so BM=2.5.