Given above is a triangle within which there are three smaller triangles of areas 2, 3 and 4 as indicated. Find the area of the remaining quadrilateral.
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Labelled the figure as above. Split the quadrilateral into two triangles with area a and b . Since △ C M P and △ C B P are sharing part of a base and having the same height. The ratio of their areas [ C B P ] [ C M P ] = B P M P = 2 1 . Similarly, [ C B P ] [ B N P ] = C P N P = 4 3 . Also
⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ [ A B P ] [ A M P ] = 2 1 [ A C P ] [ A N P ] = 4 3 ⟹ a + 3 b = 2 1 ⟹ b + 2 a = 4 3 ⟹ 2 b = a + 3 ⟹ 4 a = 3 b + 6 . . . ( 1 ) . . . ( 2 )
From ( 1 ) : a = 2 b − 3 and ( 2 ) : 4 ( 2 b − 3 ) = 3 b + 6 ⟹ b = 3 . 6 and a = 4 . 2 . Therefore, [ A M P N ] = a + b = 7 . 8 .