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I believe there is some typing mistake some where... I think that: you should mention that: AF : FE = 3 : 2 then it is simpler..

Let AF = 3 x and FE = 2 x

Draw a perpendicular from F on to AB to meet that at G. Use similar triangles principles.

EC / EF = FG / FA EC / 2x = 12 / 3x => EC = 8 cm

So : BE = 12 + 8 = 20 cm and AB = 20 cm. So AE = 20√2 cm

FE = AE * 2/5 = 8√2 cm ==================================

In case it is not a mistake and it is only AF : FC = 3 : 2 then,

FG = BC = 12 cm AG = AB - BG = AB - FC = 20 - 2 x cm

Pythagoras theorem in triangle AFG, we have: 9 x² = 12² + (20 - 2x)² => x² + 16 x - 108.8 = 0 => x = √172.8 - 8 cm ≈ 5.145 cm

AF = 3x ≈ 15.435 cm AG = 20 - 2x = 9.71 cm FC = 2x = 10.29 cm