As shown in the figure, in the trapezoidal ABCD, ad ‖ BC, ab = DC, m and N are the midpoint of AD and BC respectively, ad = 3, BC = 9, ∠ B = 45 °. Find the length of Mn

As shown in the figure, in the trapezoidal ABCD, ad ‖ BC, ab = DC, m and N are the midpoint of AD and BC respectively, ad = 3, BC = 9, ∠ B = 45 °. Find the length of Mn

Through point m, make me ‖ AB, intersect BC at point E, intersect BC at point F, ∵ ad ‖ BC, ∵ quadrilateral Abem and quadrilateral DCFM are parallelogram, ∵ be = am, CF = DM, me = AB, MF = DC, ∵ EF = bc-be-cf = bc-am-dm = BC-AD = 9-3 = 6, ∵ AB = DC, ∵ me = MF, ∵ B = 45 degree, ∵ MEF = ∠ MFE = ∠ B = 45 degree, ∵ EMF = 90 degree, ∵ m and N are the midpoint of AD and BC respectively, ∵ am = DM, BN = CN, ∴EN=FN，∴MN=12EF=3．