To fin the length of the sides of rectangle ABCD, we will be using the distance formula.
[tex]d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}[/tex]Let's first identify the coordinate of each point.
A (-4, 2)
B (-2, 4)
C (4, -2)
D (2, -4)
The coordinates will be used to substitute for the x's and th y's in the distance formula.
The length of AB is calculated as:
[tex]\begin{gathered} AB=\sqrt{[-4-(-2)]^2+(2-4)^2} \\ AB=\sqrt{(-2)^2+(-2)^2} \\ AB=\sqrt{4+4} \\ AB=\sqrt{8} \\ AB=2\sqrt{2} \end{gathered}[/tex]Meanwhile, BC's length is:
[tex]\begin{gathered} BC=\sqrt{(-2-4)^2+[4-(-2)]^2} \\ BC=\sqrt{(-6)^2+6^2} \\ BC=\sqrt{36+36} \\ BC=\sqrt{72} \\ BC=6\sqrt{2} \end{gathered}[/tex]We follow the same steps in calculating hte lengths of CD and AD:
[tex]\begin{gathered} CD=\sqrt{(4-2)^2+[-2-(-4)]^2} \\ CD=\sqrt{2^2+2^2} \\ CD=\sqrt{4+4} \\ CD=\sqrt{8} \\ CD=2\sqrt{2} \end{gathered}[/tex][tex]\begin{gathered} AD=\sqrt{(-4-2)^2+[2-(-4)]^2} \\ AD=\sqrt{(-6)^2+6^2} \\ AD=\sqrt{36+36} \\ AD=\sqrt{72} \\ AD=6\sqrt{2} \end{gathered}[/tex]The lengths of the sides of rectangle ABCD are:
AB = 2√2
BC = 6√2
CD = 2√2
AD = 6√2
The area of a recatangle is equal to the product of its length and width.
[tex]\begin{gathered} Area=L\times W \\ Area=(6\sqrt{2})(2\sqrt{2}) \\ Area=12(2) \\ Area=24units^2 \end{gathered}[/tex]The area of rectangle ABCD is 24 square units.