Question

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

Explanation:

Step1: Calculate the slopes of the sides

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

Step2: Calculate the lengths of the sides

The distance formula is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
Length of $AB$: $d_{AB}=\sqrt{(0 - a)^2+(b - 0)^2}=\sqrt{a^{2}+b^{2}}$
Length of $BC$: $d_{BC}=\sqrt{(a - 0)^2+(2b - b)^2}=\sqrt{a^{2}+b^{2}}$
Since adjacent sides are equal in length (i.e., $d_{AB}=d_{BC}$) and opposite - sides are parallel, the figure is a rhombus.
We can also check the slopes of the diagonals.
Slope of $AC$: $m_{AC}=\frac{2b-0}{a - a}$, which is undefined (vertical line).
Slope of $BD$: $m_{BD}=\frac{b - b}{2a-0}=0$ (horizontal line).
The diagonals are perpendicular, which is a property of a rhombus.
We can check if it is a square by seeing if adjacent sides are perpendicular. Since $m_{AB}\times m_{BC}=-\frac{b}{a}\times\frac{b}{a}
eq - 1$ (except when $a = b$ which is not given as a general condition), it is not a square.

Answer:

A. Rhombus