Question

what is the most specific name for the figure? a(-4a, -2b) b(-8a, -2b) c(-4a, b) d(0, b) square rhombus parallelogram rectangle

Explanation:

Step1: Calculate slope of sides

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.
Slope of $AB$: $m_{AB}=\frac{-2b+2b}{-8a + 4a}=0$.
Slope of $BC$: $m_{BC}=\frac{b + 2b}{-4a+8a}=\frac{3b}{4a}$.
Slope of $CD$: $m_{CD}=\frac{b - b}{0 + 4a}=0$.
Slope of $DA$: $m_{DA}=\frac{b + 2b}{0 + 4a}=\frac{3b}{4a}$.
Since $m_{AB}=m_{CD}$ and $m_{BC}=m_{DA}$, opposite - sides are parallel, so it is a parallelogram.

Step2: Calculate lengths of sides

Length formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Length of $AB$: $d_{AB}=\sqrt{(-8a + 4a)^2+(-2b + 2b)^2}=4|a|$.
Length of $BC$: $d_{BC}=\sqrt{(-4a + 8a)^2+(b + 2b)^2}=\sqrt{16a^{2}+9b^{2}}$.
Since $d_{AB}
eq d_{BC}$, it is not a square or a rhombus.

Step3: Check for right - angles

The product of slopes of two perpendicular lines is $- 1$.
$m_{AB}\times m_{BC}=0\times\frac{3b}{4a}=0
eq - 1$, so it is not a rectangle.

Answer:

C. Parallelogram