Question

3. The coordinates of the vertices of quadrilateral ABCD are A(0,4) , B(4,2) , C(2,-1) D(-2,1). Jorge states that quadrilateral ABCD is a parallelogram. Prove or disprove Jorge’s statement.

Answer 1

Jorge is correct — ABCD is a parallelogram.

Method 1 (vectors / opposite sides):
Compute the side vectors:
- AB = B − A = (4,2) − (0,4) = (4, −2)
- BC = C − B = (2,−1) − (4,2) = (−2, −3)
- CD = D − C = (−2,1) − (2,−1) = (−4, 2)
- DA = A − D = (0,4) − (−2,1) = (2, 3)

We see AB = (4, −2) and DC = C − D = (4, −2), so AB = DC; and BC = (−2, −3) and AD = D − A = (−2, −3), so BC = AD. Each pair of opposite sides are equal (and hence parallel), so ABCD is a parallelogram.

Method 2 (diagonals):
Diagonals AC and BD have midpoints
- Midpoint of AC = ((0+2)/2, (4+(−1))/2) = (1, 1.5)
- Midpoint of BD = ((4+(−2))/2, (2+1)/2) = (1, 1.5)

The diagonals bisect each other, which is another characterization of parallelograms. Thus ABCD is a parallelogram.