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?

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!