Question

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. (1 point) P(); R()

Answer 1

To find the coordinates of vertices P and R, we can note that a rectangle has opposite sides that are equal in length and parallel to each other.

Given that Q(4, -2) and S(-1, 3) are the vertices of the rectangle, we can find the lengths of the sides and the slopes of the sides.

The length of side QS can be found using the distance formula:
d(QS) = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d(QS) = sqrt((-1 - 4)^2 + (3 - (-2))^2)
d(QS) = sqrt((-5)^2 + (5)^2)
d(QS) = sqrt(25 + 25)
d(QS) = sqrt(50)
d(QS) = 5√2

Since QR is also equal in length, we can use this information to find the coordinates of P and R.

Starting at point Q(4, -2), we know that moving 5 units along the x-axis and 5 units along the y-axis will bring us to the point P.
P(x, y) = Q + 5(a, b)
P(x, y) = (4, -2) + 5(1, 1)
P(x, y) = (4 + 5, -2 + 5)
P(x, y) = (9, 3)

Similarly, starting at point S(-1, 3), we know that moving 5 units along the x-axis and 5 units along the y-axis will bring us to the point R.
R(x, y) = S + 5(a, b)
R(x, y) = (-1, 3) + 5(1, 1)
R(x, y) = (-1 + 5, 3 - 5)
R(x, y) = (4, -2)

Therefore, the coordinates of vertices P and R are:
P(9, 3) and R(4, -2)