Centroid Calculation

Find the centroid of the composite shape

Solution

Areas of individual parts

\[ A_1 = 4 \times 12 = 48, \quad A_2 = 6 \times 8 = 48. \]

Centroid coordinates of each part

For part 1: \[ \bar{x}_1 = 0, \quad \bar{y}_1 = 8 + \frac{4}{2} = 10. \] For part 2: \[ \bar{x}_2 = 0, \quad \bar{y}_2 = \frac{8}{2} = 4. \]

Total area

\[ A = A_1 + A_2 = 48 + 48 = 96. \]

Compute centroid coordinates

\[ \bar{x} = 0, \quad \bar{y} = \frac{A_1 \bar{y}_1 + A_2 \bar{y}_2}{A} = \frac{48(10) + 48(4)}{96} = 7. \]

Final result: \[ \boxed{\; (\bar{x}, \bar{y}) = (0, 7) \; }. \]


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