A park is in the shape of a triangle. It is plotted on a coordinate plane, and its vertices labeled Q, R, and S.

Question

Q = (1,8)
R = (-4,2)
S = (3,-1)

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, we need to determine the centroid of the triangle formed by the vertices Q, R, and S. The centroid (also referred to as the geometric center) of a triangle is the point where the three medians intersect, and it can be calculated using the average of the x-coordinates and the average of the y-coordinates of the vertices.

The formula for the centroid (C) of a triangle with vertices $A(x_1,y_1)$, $B(x_2,y_2)$, and $C(x_3,y_3)$ is given by:

$$C_x=\frac{x_1+x_2+x_3}{3}$$ $$C_y=\frac{y_1+y_2+y_3}{3}$$

Substituting the coordinates of the vertices Q, R, and S into the formulas:

• For vertex Q = (1, 8): $x_1=1$, $y_1=8$
• For vertex R = (-4, 2): $x_2=-4$, $y_2=2$
• For vertex S = (3, -1): $x_3=3$, $y_3=-1$

Now, we can calculate the x-coordinate of the centroid:

$$C_x=\frac{x_1+x_2+x_3}{3}=\frac{1+(-4)+3}{3}=\frac{1-4+3}{3}=\frac{0}{3}=0$$

Next, we calculate the y-coordinate of the centroid:

$$C_y=\frac{y_1+y_2+y_3}{3}=\frac{8+2+(-1)}{3}=\frac{8+2-1}{3}=\frac{9}{3}=3$$

Thus, the coordinates of the centroid, where the information center should be placed, are:

$$C=(C_x,C_y)=(0,3)$$

Therefore, the information center should be placed at the coordinate (0, 3). This location is the average of the three vertices and serves as the center point within the triangular park.