Find the centroid of the common region used in engineering F

Find the centroid of the common region used in engineering. Find the centroid of the parallelogram with vertices (0, 0), (a, 0), (b, c), and (a + b, c) (see figure).

Solution

Given that

Vertices of parllelogram are ,

let ( x₁ , y₁) = ( 0 , 0 )

(x₂ , y₂ ) = ( a , 0 )

( x₃ , y₃ ) = ( b , c )

( x₄ , y₄ ) = ( a+b , c )

Centroid of parllelogram ( x ,   ) = [ ( x₁ + x₂ + x₃ + x₄)/4 , (y₁ + y₂ + y₃ + y₄)/4 ]

= [ ( 0 + a + b + (a+b) )/4 , ( 0 + 0 + c + c )/4 ]

= ( ( 2a + 2b)/4 , 2c/4 )

= ( 2(a+b)/4 , 2c/4 )

= ( (a + b)/2 , c/2 )

Therefore ,

Centroid of parllelogram = ( (a + b)/2 , c/2 )