https://file.aiquickdraw.com/mathbot/file/17257194784856ub3whng.jpg

To determine the location of the centroid for the given shape, we can break it down into simpler geometric shapes and find the centroid for each. The figure appears to consist of three distinct parts: 

1. **Top Triangle** (equilateral with a base of 150 mm and height of 100 mm)
2. **Middle Rectangle** (75 mm wide and 250 mm tall)
3. **Bottom Rectangle** (300 mm wide and 150 mm tall)

### Steps to Find the Centroid:

1. **Calculate Areas:**
   - **Top Triangle:** 
     \[
     A_1 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 150 \, \text{mm} \times 100 \, \text{mm} = 7500 \, \text{mm}^2
     \]
   - **Middle Rectangle:** 
     \[
     A_2 = \text{width} \times \text{height} = 75 \, \text{mm} \times 250 \, \text{mm} = 18750 \, \text{mm}^2
     \]
   - **Bottom Rectangle:** 
     \[
     A_3 = \text{width} \times \text{height} = 300 \, \text{mm} \times 150 \, \text{mm} = 45000 \, \text{mm}^2
     \]

2. **Determine Centroid Locations** (relative to the bottom of the figure):
   - **Top Triangle:** The centroid of a triangle is located \( \frac{1}{3} \) of the height from its base.
     \[
     y_1 = 500 \, \text{mm} - \frac{1}{3} \times 100 \, \text{mm} = 466.67 \, \text{mm}
     \]
   - **Middle Rectangle:** The centroid of a rectangle is located halfway through its height.
     \[
     y_2 = 250 \, \text{mm} + \frac{250}{2} = 375 \, \text{mm}
     \]
   - **Bottom Rectangle:** The centroid of a rectangle is located halfway through its height.
     \[
     y_3 = \frac{150}{2} = 75 \, \text{mm}
     \]

3. **Compute the Total Area:**
   \[
   A_{\text{total}} = A_1 + A_2 + A_3 = 7500 + 18750 + 45000 = 71250 \, \text{mm}^2
   \]

4. **Calculate the Centroid's Y-Coordinate:**
   Use the weighted average formula to determine the centroid location in the y-direction.
   \[
   y_{\text{centroid}} = \frac{A_1 y_1 + A_2 y_2 + A_3 y_3}{A_{\text{total}}}
   \]
   Substituting values:
   \[
   y_{\text{centroid}} = \frac{7500 \times 466.67 + 18750 \times 375 + 45000 \times 75}{71250}
   \]
   Let's compute this value.

The centroid of the given composite shape is located approximately **195.18 mm** from the bottom of the figure in the vertical direction.

Would you like more details or have any questions? Here are some related questions:

1. How do we find the centroid for irregular shapes?
2. How does the location of the centroid affect the stability of objects?
3. Can the centroid be outside the physical boundaries of a shape?
4. How does the shape's symmetry influence the centroid's position?
5. What is the significance of the centroid in structural engineering?

**Tip:** When determining centroids for composite shapes, always break the shape into simpler parts and calculate their individual centroids first.

Determine the location of the centroid of the shape shown in the image.

Learn how to find the centroid of a complex geometric shape by decomposing it into simpler parts such as rectangles and triangles. This guide provides a step-by-step approach to calculating areas and centroid locations for each part, using basic geometry formulas.

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