Question

Prove that the points \(\mathrm{A}(-2, -1), \mathrm{B}(1, 0), \mathrm{C}(4, 3)\) and \(\mathrm{D}(1, 2)\) are the vertices of a parallelogram ABCD.

Answer 1

Given: Points \(A(-2, -1)\), \(B(1, 0)\), \(C(4, 3)\) and \(D(1, 2)\).

To prove: ABCD is a parallelogram.

Proof:

1. In a parallelogram, the diagonals bisect each other. Conversely, if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
2. Mid-point of diagonal AC: $$\left( \frac{-2 + 4}{2}, \frac{-1 + 3}{2} \right) = (1, 1)$$
3. Mid-point of diagonal BD: $$\left( \frac{1 + 1}{2}, \frac{0 + 2}{2} \right) = (1, 1)$$
4. The mid-points of AC and BD are the same, so the diagonals AC and BD bisect each other.
5. Therefore, ABCD is a parallelogram.

Hence proved.