Question

A park is in the shape of a triangle. It is plotted on a coordinate plane, and its vertices labeled Q, R, and S. An information center is to be placed at the center of the park. Where should the information center be placed? Show your work and explain how you know. (2 points)

Q is 1, 8
R is -4, 2
S is 3, 1

Answer 1

To find the position of the information center in the triangular park formed by the points Q, R, and S, we should calculate the centroid of the triangle. The centroid is the point where the three medians of the triangle intersect and serves as the center of mass, which is a suitable location for the information center.

Given the vertices of the triangle:
- \( Q (1, 8) \)
- \( R (-4, 2) \)
- \( S (3, 1) \)

The formula for finding the centroid \( (G_x, G_y) \) of a triangle given its vertices \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is:

\[
G_x = \frac{x_1 + x_2 + x_3}{3}
\]
\[
G_y = \frac{y_1 + y_2 + y_3}{3}
\]

Now we can plug in the coordinates of points Q, R, and S into the formulas:

1. Calculate \( G_x \):

\[
G_x = \frac{1 + (-4) + 3}{3} = \frac{1 - 4 + 3}{3} = \frac{0}{3} = 0
\]

2. Calculate \( G_y \):

\[
G_y = \frac{8 + 2 + 1}{3} = \frac{11}{3} \approx 3.67
\]

Thus, the coordinates of the centroid are:

\[
G(0, \frac{11}{3}) \quad \text{or approximately} \quad G(0, 3.67)
\]

### Conclusion:
The information center should be placed at the coordinates \( G(0, \frac{11}{3}) \) or approximately \( (0, 3.67) \). This point represents the centroid of the triangular park and serves as the optimal point for placing the information center, as it is centrally located within the triangle defined by points Q, R, and S.