Determine whether parallelogram JKLM with vertices J(-7, -2), K(0, 4), L(9, 2), and M(2, -4) is a rhombus, square, rectangle, or all three. Provide a coordinate proof to justify your answer.

Question

Determine whether parallelogram JKLM with vertices J(-7, -2), K(0, 4), L(9, 2), and M(2, -4) is a rhombus, square, rectangle, or all three. Provide a coordinate proof to justify your answer. 

Answer 1

To determine if this parallelogram is a rhombus, square, rectangle, or all three, we must determine the lengths and slopes of each side of the parallelogram.

(1) Find the slope of each side:

Use the general formula to find the slope: m = (y₂-y₁)/(x₂-x₁)

JK: m = (4 - (-2))/(0 - (-7)) = 6/7

LM: m = (-4 - 2)/(2 - 9) = 6/7

KL: m = (2 - 4)/(9 - 0) = -2/9

ML: m = (-2 - (-4))/(-7 - 2) = -2/9

Line segments JK and LM are opposite sides and have the same slope. Line segments KL and ML are opposite sides and have the same slope. Therefore, line segments JK and LM are parallel to each other, and line segments KL and ML are parallel to each other.

Notice how the slopes of the adjacent sides do not multiply to -1 (6/7*-2/9 = -12/63). This means they are not perpendicular; therefore, this parallelogram is not a rectangle.

(2) Find the lengths of each side:

Use the distance formula to find the length of each side: d = √[(x₂-x₁)² + (y₂-y₁)²]

JK: m = √[(0 - (-7))² + (4 - (-2))²] = √(85)

LM: m = √[(2 - 9)² + (-4 - 2)²] = √(85)

KL: m = √[(9 - 0)² + (2 - 4)²] = √(85)

ML: m = √[(-7 - 2)² + (-2 - (-4))²] = √(85)

Notice how each of the sides of the parallelogram are the same length. This means that this parallelogram is also a rhombus. It cannot also be a square, since the interior angles are not right angles.

Therefore, the parallelogram JKLM is a rhombus.