a) Theo giả thiết: MN // BC nên \(\frac = \frac = \frac\)

Mà \(AK = \frac{1}{3} \Rightarrow \frac = \frac{1}{3}\)

Lại có: EF // BC nên \(\frac = \frac = \frac = \frac{2}{3}\)

b) Ta có: S_ABC = \(\frac{1}{2}.AH.BC\)

S_AMN = \(\frac{1}{2}.AK.MN = \frac{1}{2}.\frac{1}{3}AH.\frac{1}{3}BC = \frac{1}{9}{S_{ABC}}\)

S_AEF = \(\frac{1}{2}AI.EF = \frac{1}{2}.\frac{2}{3}.AH.\frac{2}{3}BE = \frac{4}{9}{S_{ABC}}\)

S_MNFE = S_AFE – S_AMN = \(\frac{4}{9}{S_{ABC}} - \frac{1}{9}{S_{ABC}} = \frac{1}{3}{S_{ABC}} = \frac{1}{3}.90 = 30\left( {c{m^2}} \right)\)

Vậy S_MNFE = 30cm².