Question

what is the most specific name for the figure? p(0, 0) q(0, 2a) r(2a, 2a) s(2a, 0)

Explanation:

Step1: Calculate side - lengths

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $PQ$ with $P(0,0)$ and $Q(0,2a)$: $d_{PQ}=\sqrt{(0 - 0)^2+(2a-0)^2}=2a$.
For side $QR$ with $Q(0,2a)$ and $R(2a,2a)$: $d_{QR}=\sqrt{(2a - 0)^2+(2a - 2a)^2}=2a$.
For side $RS$ with $R(2a,2a)$ and $S(2a,0)$: $d_{RS}=\sqrt{(2a - 2a)^2+(0 - 2a)^2}=2a$.
For side $SP$ with $S(2a,0)$ and $P(0,0)$: $d_{SP}=\sqrt{(0 - 2a)^2+(0 - 0)^2}=2a$.

Step2: Check angles

The slope of $PQ$ is undefined ($x_1=x_2 = 0$), the slope of $QR$ is $m_{QR}=\frac{2a - 2a}{2a-0}=0$. Since the product of the slopes of two - adjacent sides $PQ$ and $QR$ is $0\times$ (undefined) which implies they are perpendicular. Similarly, other adjacent sides are perpendicular.

Answer:

Square