Question

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

Explanation:

Step1: Calculate side - lengths using distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $AB$ with $A(0,-2a)$ and $B(-3a,-2a)$:
$AB=\sqrt{(-3a - 0)^2+(-2a+2a)^2}=\sqrt{(-3a)^2}=3|a|$.
For side $BC$ with $B(-3a,-2a)$ and $C(-3a,a)$:
$BC=\sqrt{(-3a + 3a)^2+(a + 2a)^2}=\sqrt{(3a)^2}=3|a|$.
For side $CD$ with $C(-3a,a)$ and $D(0,a)$:
$CD=\sqrt{(0 + 3a)^2+(a - a)^2}=\sqrt{(3a)^2}=3|a|$.
For side $DA$ with $D(0,a)$ and $A(0,-2a)$:
$DA=\sqrt{(0 - 0)^2+(-2a - a)^2}=\sqrt{(-3a)^2}=3|a|$.

Step2: Check angle - properties

The slope of $AB$: $m_{AB}=\frac{-2a+2a}{-3a - 0}=0$.
The slope of $BC$: $m_{BC}=\frac{a + 2a}{-3a+3a}$ is undefined, so $AB\perp BC$.
Since all sides are equal ($AB = BC=CD = DA$) and one - angle is a right - angle ($\angle ABC = 90^{\circ}$), the figure is a square.

Answer:

A. Square