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To find the centroid of the triangle \( QRS \) shown on the coordinate grid without using the centroid formula, we can calculate it step by step. The centroid of a triangle is the point where the three medians intersect, and it is the average of the vertices' coordinates.

### Step 1: Identify the coordinates of the vertices
From the graph, we can visually estimate the coordinates of the triangle's vertices \( Q \), \( R \), and \( S \):
- \( Q \) appears to be at \( (-2, 2) \).
- \( R \) appears to be at \( (3, 8) \).
- \( S \) appears to be at \( (6, 2) \).

### Step 2: Average the coordinates of the vertices
The coordinates of the centroid \( C \) are the averages of the x-coordinates and the y-coordinates of \( Q \), \( R \), and \( S \).

- For the x-coordinate of the centroid:
  \[
  x_C = \frac{x_Q + x_R + x_S}{3} = \frac{-2 + 3 + 6}{3} = \frac{7}{3} \approx 2.33
  \]
  
- For the y-coordinate of the centroid:
  \[
  y_C = \frac{y_Q + y_R + y_S}{3} = \frac{2 + 8 + 2}{3} = \frac{12}{3} = 4
  \]

### Step 3: Conclusion
Thus, the centroid \( C \) of the triangle is approximately \( \left( \frac{7}{3}, 4 \right) \) or \( (2.33, 4) \).

Would you like further details or have any other questions?

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Here are 5 related questions to expand on the topic:
1. How do you derive the formula for the centroid of a triangle?
2. What are the properties of a centroid in a triangle?
3. How does the centroid divide each median of the triangle?
4. What are the differences between a centroid and other triangle centers (orthocenter, circumcenter, incenter)?
5. Can the centroid ever lie outside of the triangle? Why or why not?

**Tip:** The centroid of a triangle always lies inside the triangle, regardless of its shape or size!

Without using the centroid formula, find the centroid of the triangle given above.

This problem explores how to find the centroid of a triangle on a coordinate grid without using the centroid formula. Key concepts include coordinate geometry and the properties of a triangle's centroid. Suitable for students in Grades 8-10.

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