To determine the centroid of the given composite shape, let's analyze each part in the figure.

Steps to Solve

1. Identify each section (A, B, and C):
   From the figure, it seems that section:

   • A is the horizontal rectangle at the top,
   • B is the vertical rectangle (central part),
   • C is the horizontal rectangle at the bottom.

2. Calculate the area $A_i$ for each section:

   • Section A: Length = 12 cm, Height = 2 cm. $A_A = 12 \times 2 = 24 \, \text{cm}^2$
   • Section B: Length = 2 cm, Height = 12 cm. $A_B = 2 \times 12 = 24 \, \text{cm}^2$
   • Section C: Length = 8 cm, Height = 2 cm. $A_C = 8 \times 2 = 16 \, \text{cm}^2$

3. Find the centroid coordinates $\bar{x}_i$ and $\bar{y}_i$ for each section:

   • Section A: The center is located at: $\bar{x}_A = \frac{12}{2} = 6 \, \text{cm}, \quad \bar{y}_A = 12 + 1 = 13 \, \text{cm}$
   • Section B: The center is located at: $\bar{x}_B = \frac{2}{2} = 1 \, \text{cm}, \quad \bar{y}_B = \frac{12}{2} = 6 \, \text{cm}$
   • Section C: The center is located at: $\bar{x}_C = \frac{8}{2} = 4 \, \text{cm}, \quad \bar{y}_C = 1 \, \text{cm}$

4. Calculate $A_i \cdot \bar{x}_i$ and $A_i \cdot \bar{y}_i$ for each section:

   • Section A: $A_A \cdot \bar{x}_A = 24 \times 6 = 144$ $A_A \cdot \bar{y}_A = 24 \times 13 = 312$
   • Section B: $A_B \cdot \bar{x}_B = 24 \times 1 = 24$ $A_B \cdot \bar{y}_B = 24 \times 6 = 144$
   • Section C: $A_C \cdot \bar{x}_C = 16 \times 4 = 64$ $A_C \cdot \bar{y}_C = 16 \times 1 = 16$

5. Sum all areas and the products $A_i \cdot \bar{x}_i$ and $A_i \cdot \bar{y}_i$:

   • Total Area $\sum A_i = 24 + 24 + 16 = 64 \, \text{cm}^2$
   • $\sum A_i \cdot \bar{x}_i = 144 + 24 + 64 = 232$
   • $\sum A_i \cdot \bar{y}_i = 312 + 144 + 16 = 472$

6. Calculate the centroid coordinates $\bar{x}$ and $\bar{y}$: $\bar{x} = \frac{\sum A_i \cdot \bar{x}_i}{\sum A_i} = \frac{232}{64} = 3.625 \, \text{cm}$ $\bar{y} = \frac{\sum A_i \cdot \bar{y}_i}{\sum A_i} = \frac{472}{64} = 7.375 \, \text{cm}$

Final Answer

The centroid of the composite shape is located at: $(\bar{x}, \bar{y}) = (3.625 \, \text{cm}, 7.375 \, \text{cm})$

Would you like a detailed explanation of each calculation step, or do you have any questions?

Follow-up Questions

1. What is the purpose of finding the centroid in this context?
2. How would the centroid change if section dimensions were altered?
3. What other methods can be used to find the centroid of complex shapes?
4. Why is the centroid located at the calculated coordinates, and what does it imply about the shape’s balance?
5. How can this method apply to 3-dimensional shapes?

Tip:

Remember, when calculating centroids for composite shapes, always break down the figure into simpler geometric parts.