Question

A park in the square of a triangle. It is potted 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

Answer 1

To find the location for the information center to be placed at the center of the park, which is in the shape of a triangle with vertices \( Q \), \( R \), and \( S \) on a coordinate plane, we can determine the centroid of the triangle.

The centroid (or geometric center) of a triangle is the average of the coordinates of its vertices. The formula for finding the coordinates of the centroid \( (G_x, G_y) \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:

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

### Steps to Find the Centroid

1. **Identify the coordinates of the vertices**:
- Let the coordinates of the vertices be:
- \( Q (x_1, y_1) \)
- \( R (x_2, y_2) \)
- \( S (x_3, y_3) \)

2. **Substitute the coordinates into the centroid formula**:
\[
G_x = \frac{x_1 + x_2 + x_3}{3}
\]
\[
G_y = \frac{y_1 + y_2 + y_3}{3}
\]

3. **Calculate \( G_x \) and \( G_y \)** to find the coordinates of the centroid \( G \).

### Example

Let's say the vertices are as follows:
- \( Q(1, 2) \)
- \( R(4, 6) \)
- \( S(7, 2) \)

Calculate the centroid:

1. **Coordinates of \( Q, R, S \)**:
- \( x_1 = 1, y_1 = 2 \)
- \( x_2 = 4, y_2 = 6 \)
- \( x_3 = 7, y_3 = 2 \)

2. **Applying the formula**:
\[
G_x = \frac{1 + 4 + 7}{3} = \frac{12}{3} = 4
\]
\[
G_y = \frac{2 + 6 + 2}{3} = \frac{10}{3} \approx 3.33
\]

### Conclusion

The coordinates of the centroid (the recommended location for the information center) would be approximately \( (4, \frac{10}{3}) \) or \( (4, 3.33) \).

Thus, the information center should be placed at the point \( \boxed{(4, \frac{10}{3})} \) based on the example coordinates provided. You can substitute the actual coordinates of your triangle's vertices to find the precise location in your scenario.