Question

Determine whatever ABCD is a parallelogram by using the midpoint formula.

A(3,3) B (1.2) C(-3.1) D (-1.4)

Answer 1

Not a parallelogram.

Explanation:

By the midpoint formula, the midpoint of a segment between
`(x_(0),\, y_(0))` and
`(x_(1),\, y_(1))` is:

`\begin{aligned} \left((x_(0) + x_(1))/(2),\, (y_(0) + y_(1))/(2)\right)\end{aligned}`.

A quadrilateral is a parallelogram if and only if the midpoints of the two diagonals are the same.

The two diagonals of quadrilateral
`{\sf ABCD}` are segment
`\sf{AC}` and segment
`{\sf BD}`, respectively.

Using the midpoint formula, the midpoint of segment
`{\sf AC}` (between
`{\sf A}\; (3,\, 3)` and
`{\sf C}\; (-3,\, 1)`) would be:

`\begin{aligned} & \left((3 + (-3))/(2),\, (3 + 1)/(2)\right) \\ =\; & (0,\, 2)\end{aligned}`.

Likewise, the midpoint of segment
`{\sf BD}` (between
`{\sf B}\; (1,\, 2)` and
`{\sf D}\; (-1,\, 4)` would be
`(0,\, 3)`.

Thus, quadrilateral
`{\sf ABCD}` would not be a parallelogram since the midpoints of its two diagonals are not the same.