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To determine the center of gravity (CG) of the shaded area in the given diagram, we must break the area into simpler geometrical shapes and use the formula for the centroid of composite areas.

### Steps:
1. **Divide the shaded area** into basic shapes:
   - A triangle \( \text{OAB} \)
   - A rectangle \( \text{ACD} \)
   - Subtract the area of the small triangle \( \text{OBD} \) (since it is not shaded).

2. **Centroid of individual areas**:
   - For each shape, the centroid \( (x_i, y_i) \) is determined from reference axes (in this case, the \( x \)-axis and \( y \)-axis through point \( O \)).
   
3. **Apply the centroid formula** for the composite area:

   \[
   \bar{x} = \frac{\sum (x_i \cdot A_i)}{\sum A_i}, \quad \bar{y} = \frac{\sum (y_i \cdot A_i)}{\sum A_i}
   \]

   where \( A_i \) is the area of each shape, and \( (x_i, y_i) \) are the coordinates of the centroid of each shape.

---

### Areas and Centroids:
1. **Triangle \( OAB \) (large triangle)**:
   - Area \( A_1 = \frac{1}{2} \times 40 \, \text{cm} \times 30 \, \text{cm} = 600 \, \text{cm}^2 \)
   - Centroid \( \left( x_1, y_1 \right) = \left( \frac{1}{3} \times 40, \frac{1}{3} \times 30 \right) = (13.33, 10) \)

2. **Rectangle \( ACD \)**:
   - Area \( A_2 = 20 \, \text{cm} \times 20 \, \text{cm} = 400 \, \text{cm}^2 \)
   - Centroid \( \left( x_2, y_2 \right) = \left( 40 - \frac{20}{2}, \frac{20}{2} \right) = (30, 10) \)

3. **Small triangle \( OBD \) (subtracted area)**:
   - Area \( A_3 = \frac{1}{2} \times 20 \, \text{cm} \times 20 \, \text{cm} = 200 \, \text{cm}^2 \)
   - Centroid \( \left( x_3, y_3 \right) = \left( \frac{1}{3} \times 20, \frac{1}{3} \times 20 \right) = (6.67, 6.67) \)

---

### Applying the Formula:
\[
\bar{x} = \frac{(13.33 \times 600) + (30 \times 400) - (6.67 \times 200)}{600 + 400 - 200}
\]
\[
\bar{y} = \frac{(10 \times 600) + (10 \times 400) - (6.67 \times 200)}{600 + 400 - 200}
\]

Solving these will give us the coordinates of the centroid \( (\bar{x}, \bar{y}) \) for the shaded area.

Would you like a detailed calculation of these values? Also, here are five related questions to expand on this topic:

1. How does the centroid shift if the dimensions of the shapes change?
2. What happens to the centroid if the shapes are mirrored about the x-axis?
3. Can this approach be used to find the centroid of any irregular shape?
4. How does the center of gravity differ from the centroid in three-dimensional objects?
5. What are the common applications of centroids in structural engineering?

**Tip:** Always break complex shapes into simpler ones when calculating centroids or moments of inertia for accuracy.

Learn how to find the center of gravity (CG) of a composite shaded area by breaking it down into basic shapes. This step-by-step solution uses the centroid formula for triangles and rectangles, suitable for students in Grades 10-12.

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