Question

what is the most specific name for the figure? a(-a, 0) b(-3a, b) c(-2a, 2b) d(0, b)

Explanation:

Step1: Calculate the slope of sides

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Slope of $AB$: $m_{AB}=\frac{b - 0}{-3a-(-a)}=\frac{b}{-2a}=-\frac{b}{2a}$.
Slope of $BC$: $m_{BC}=\frac{2b - b}{-2a-(-3a)}=\frac{b}{a}$.
Slope of $CD$: $m_{CD}=\frac{b - 2b}{0-(-2a)}=-\frac{b}{2a}$.
Slope of $DA$: $m_{DA}=\frac{0 - b}{-a - 0}=\frac{b}{a}$.

Step2: Check parallel - side condition

Since $m_{AB}=m_{CD}$ and $m_{BC}=m_{DA}$, opposite sides are parallel, so it is a parallelogram.

Step3: Check perpendicular - side condition

$m_{AB}\times m_{BC}=-\frac{b}{2a}\times\frac{b}{a}=-\frac{b^{2}}{2a^{2}}
eq - 1$, so adjacent sides are not perpendicular, it is not a rectangle, square.
We don't have information to show all sides are equal (for rhombus).

Answer:

Parallelogram