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.