Question

The coordinates of the vertices of quadrilateral ABCD are A(-3,2), B (-4,-2), C(4,2), D(3,2). Justin states that quadrilateral ABCD is a parallelogram. Is Justin correct?

A) Yes
B) No

Explain your reasoning (2 points). Show your work (2 points).

Answer 1

After calculating the lengths of the sides of the quadrilateral ABCD using the distance formula, it is evident that opposite sides are neither congruent nor equal in length, meaning quadrilateral ABCD is not a parallelogram, and Justin is incorrect.

Step-by-step explanation:

To determine if quadrilateral ABCD with vertices A(-3,2), B (-4,-2), C(4,2), and D(3,2) is a parallelogram, we must check if the opposite sides are congruent and parallel. We can calculate the lengths of the sides using the distance formula, which is √((x2-x1)² + (y2-y1)²).

First, we calculate the distance AB: √((-4+3)² + (-2-2)²) = √((-1)² + (-4)²) = √(1 + 16) = √17.

Next, we calculate the distance BC: √((4+4)² + (2+2)²) = √(8² + 4²) = √(64 + 16) = √80.

Then we calculate the distance CD: √((3-4)² + (2-2)²) = √((-1)² + 0²) = √1.

Finally, we calculate the distance DA: √((-3-3)² + (2-2)²) = √((-6)² + 0²) = √36.

We now see that AB (√17) is not equal to CD (√1), and BC (√80) is not equal to DA (√36). Since opposite sides are neither congruent nor equal in length, we can conclude that quadrilateral ABCD is not a parallelogram. Therefore, Justin is incorrect.