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Each side of triangle ABC is divided into three equal parts. Find the ratio of the are of hexagon `P Q R S T U` to the area of the triangle ABC.

Monjur Alahi

Each side of triangle ABC is divided into three equal parts. Find the ratio of the are of hexagon `P Q R S T U` to the area of the triangle ABC.


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Salim Ahmed Kawsar

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Area of hexagon PQRSTU
`Delta_(1) = " area of " Delta ABC - " area of " Delta APQ - " area of " Delta BTU - " area of " DeltaCRS`
`= Delta -(1)/(2) ((b)/(3). (c)/(3) sin A) - (1)/(2) ((a)/(3).(c)/(3) sin B) - (1)/(2) ((a)/(3).(b)/(3) sin C)`
`= Delta - (Delta)/(3) = (2Delta)/(3)`
`:. (Delta_(1))/(Delta) = (2)/(3)`

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