Question

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

Answer 1

Answer:

Not a parallelogram.

Step-by-step explanation:

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

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

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:

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

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.