Question

identify the figure with the vertices q(-3,2), r(1,2), s(1,-4), and t(-3,-4)

Explanation:

Step1: Calculate side - lengths

Use the distance formula $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.
For side $QR$: $x_1=-3,y_1 = 2,x_2 = 1,y_2 = 2$. Then $QR=\sqrt{(1-(-3))^2+(2 - 2)^2}=\sqrt{4^2+0^2}=4$.
For side $RS$: $x_1 = 1,y_1 = 2,x_2 = 1,y_2=-4$. Then $RS=\sqrt{(1 - 1)^2+(-4 - 2)^2}=\sqrt{0^2+(-6)^2}=6$.
For side $ST$: $x_1 = 1,y_1=-4,x_2=-3,y_2=-4$. Then $ST=\sqrt{(-3 - 1)^2+(-4+4)^2}=\sqrt{(-4)^2+0^2}=4$.
For side $TQ$: $x_1=-3,y_1=-4,x_2=-3,y_2 = 2$. Then $TQ=\sqrt{(-3+3)^2+(2 + 4)^2}=\sqrt{0^2+6^2}=6$.

Step2: Analyze side - length and angle relationships

Opposite sides $QR$ and $ST$ are equal ($QR = ST=4$), and opposite sides $RS$ and $TQ$ are equal ($RS=TQ = 6$). Also, since the change in $x$ - values or $y$ - values for adjacent sides results in right - angled corners (e.g., for $QR$ and $RS$, one has a change in $x$ only and the other has a change in $y$ only), the figure is a rectangle.

Answer:

Rectangle