To locate the centroid of the shaded area shown in the diagram, we will calculate the centroid coordinates $(\bar{x}, \bar{y})$ using the given dimensions and geometry of the triangle.

Solution:

The centroid of a triangular area can be determined directly using formulas:

1. Centroid along $x$-axis ($\bar{x}$): $\bar{x} = \frac{1}{3}(a + b)$

   • Here, $a$ is the length of the top horizontal side (base of the smaller triangle), and $b$ is the distance from the left to the base of the larger triangle.

2. Centroid along $y$-axis ($\bar{y}$): $\bar{y} = \frac{h}{3}$

   • $h$ is the vertical height of the triangular area.

Given:

• The diagram's key measurements:
  • The top base is $a$,
  • The bottom base begins at $x = b$,
  • The height of the shaded region is $h$.

Thus, the centroid coordinates are: $\boxed{\left(\frac{1}{3}(a+b), \frac{h}{3}\right)}$

This result matches the provided centroid formula shown in the image.

Would you like a detailed derivation of this formula or additional examples to understand centroid calculations better?

Further Related Questions:

1. How can we derive the formula for a centroid of a triangle geometrically?
2. What are some real-world applications of centroid calculations in engineering or physics?
3. How would the centroid location change if the shape was not a triangle but a composite figure?
4. How do integrals help in finding centroids of irregular shapes?
5. Can you explain why the centroid's location divides the triangle into equal-area sections?

Tip:

For composite shapes, split the figure into basic geometric parts (e.g., rectangles, triangles) and find the weighted average of their centroids.