Rectangle PQRS has vertices Q (4, -2) and S (-1, 3). What are the coordinates of vertices P and R? Note, the vertex labels are located counterclockwise on the graph.

Answer 1

To find the coordinates of vertices P and R, we need to determine the coordinates of the opposite vertices of the rectangle.

Let's start with the given vertices:

Q (4, -2) and S (-1, 3)

To find the coordinates of P, we need to notice that a rectangle has opposite sides that are equal in length and parallel to each other. This means that the line joining Q and S is one of the diagonals of the rectangle, and the midpoint of this diagonal will be the center of the rectangle, which will help us locate the positions of P and R.

The midpoint of the line joining Q and S can be found by averaging the x-coordinates and y-coordinates of Q and S separately:

Midpoint:
x-coordinate: (4 - 1)/2 = 3/2 = 1.5
y-coordinate: (-2 + 3)/2 = 1/2 = 0.5

The midpoint is at (1.5, 0.5).

Since the opposite sides of a rectangle are equal in length and parallel, we can use the midpoint to find the coordinates of P and R. P and R should be the same distance away from the midpoint and form a rectangle with Q and S.

To find P:
Starting at the midpoint, we need to move the same distance as from Q to the midpoint but in the opposite direction. From the midpoint (1.5, 0.5), move 2.5 units to the left and 2.5 units down to find point P:

P (1.5 - 2.5, 0.5 - 2.5)
P (-1, -2)

To find R:
Starting at the midpoint, we need to move the same distance as from S to the midpoint but in the opposite direction. From the midpoint (1.5, 0.5), move 2.5 units to the right and 2.5 units up to find point R:

R (1.5 + 2.5, 0.5 + 2.5)
R (4, 3)

Therefore, the coordinates of vertices P and R are:
P (-1, -2)
R (4, 3)