https://file.aiquickdraw.com/mathbot/file/17264074108565rdamwf5.jpg

The image shows a geometric figure composed of a rectangle and a right triangle. We are tasked with finding the center of mass (centroid) of this composite shape. The shape is composed of:

1. A rectangle with dimensions \(8 \times 10\) units.
2. A right triangle located at the bottom-left corner with its right angle at \((0, 0)\), height of 6 units, and a base of 6 units.

### Steps to find the centroid:
We treat this as a composite area problem, breaking the shape into the following parts:
- **Rectangle**: Without the triangle, it extends from \(x = 6\) to \(x = 8\), height 10 units.
- **Right triangle**: Subtract the triangular area from the rectangle.

1. **Centroid of the Rectangle**:
   - Area: \(A_1 = 8 \times 10 = 80\)
   - The centroid is located at:
     \[
     (x_1, y_1) = \left(\frac{6 + 8}{2}, \frac{10}{2}\right) = (7, 5)
     \]

2. **Centroid of the Triangle**:
   - Area: \(A_2 = \frac{1}{2} \times 6 \times 6 = 18\)
   - The centroid of a right triangle is located at:
     \[
     (x_2, y_2) = \left(\frac{1}{3} \times 6, \frac{1}{3} \times 6\right) = (2, 2)
     \]

3. **Calculating the overall centroid**:
   The centroid of the composite figure is found using the weighted average of the centroids of the two shapes:
   \[
   x_{\text{centroid}} = \frac{x_1 A_1 - x_2 A_2}{A_1 - A_2}
   \]
   \[
   y_{\text{centroid}} = \frac{y_1 A_1 - y_2 A_2}{A_1 - A_2}
   \]

Let me calculate that for you.The centroid (center of mass) of the composite shape is located at approximately \((8.45, 5.87)\).

Would you like further details or clarification?

Here are 5 related questions to explore this topic:
1. How do we calculate centroids for more complex shapes?
2. What are some applications of centroid calculations in engineering?
3. How can we determine the centroid of a 3D object?
4. How does the centroid affect the balance of an object?
5. Can the centroid lie outside the object?

**Tip**: For composite shapes, breaking them into simpler parts and calculating each part's centroid and area helps simplify the problem.

Determine the centroid (center of mass) of the composite shape in the image.

Learn how to determine the centroid (center of mass) of a composite geometric shape, involving a rectangle and a right triangle. This problem utilizes key geometry concepts such as centroid calculation for composite shapes and the formulas for the centroid of basic shapes.

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