Question

One diagonal of a parallelogram has endpoints at P(-2, 5) and R(1, -1). The other diagonal has endpoints at Q(1,5) and S(-2,-1). What type of parallelogram is PQRS? Explain

Answer 1

Rectangle

Explanation:

It can help to plot the points on a graph.

Points P and Q have the same y-coordinates (5), so are on the same horizontal line. Points P and S have the same x-coordinates (-2), so are on the same vertical line. The sides of this parallelogram are perpendicular to each other, so it is a rectangle.

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Analytically, you can find the lengths of the diagonals. The vector PR has coordinates R-P = (1-(-2), (-1-5)) = (3, -6), so will have length √(3²+(-6)²) = 3√5. The vector QS has coordinates S-Q = (-2-1, -1-5) = (-3, -6), so will have length √((-3)²+(-6)²) = 3√5.

When the diagonals of a parallelogram have the same length, it is a rectangle. (These vectors are not at right angles to each other, so the rectangle is not a square.)