Carterville was built along a grid using a coordinate plane. City Hall was located at . The Fire Department was located at , and the Police Department was located at . Each square on the coordinate plane is equal to 1 square mile. City planning wants to build a park such that the shape created by the Fire Department, City Hall and the Police Department form a rectangle. What should the coordinates of the new park be?

Question

A.

B.

C.

D.

Answer 1

To determine the coordinates of the new park that will complete the rectangle formed by the locations of the Fire Department, City Hall, and the Police Department, we first need to identify the coordinates of each of these locations.

Let's assume:

• City Hall (C) is at coordinates \((x_1, y_1)\)
• Fire Department (F) is at coordinates \((x_2, y_2)\)
• Police Department (P) is at coordinates \((x_3, y_3)\)

For the formation of a rectangle, if we denote the unknown point (the park) as \(P'(x', y')\), it must satisfy the properties of a rectangle. This means that the diagonals must connect the opposite corners.

Using the properties of a rectangle, we know that the diagonals connect the following points:

• If we consider City Hall and the Fire Department as one pair of opposite corners, and the Police Department as one corner, we can calculate the fourth corner (the park) as follows:

\[ P'(x', y') = (x_1 + x_2 - x_3, y_1 + y_2 - y_3) \]

We'll need the specific coordinates of City Hall, the Fire Department, and the Police Department to calculate \(P'\). Please provide those coordinates so we can find the exact coordinates for the new park.