The formula of the distance between 2 points states that the distance AB between points A(a, b) and B(c, d) is found as follows:
[tex]\displaystyle{ AB= \sqrt{(a-c)^2+(b-d)^2} [/tex].
Let the points of the polygon be A(-6,-1), B(-3,-4), C(6,5) and D(3,8). Then, the perimeter of the polygon ABCD is
AB+BC+CD+DA.
We can find each of AB, BC, CD, DA by the Distance Formula as follows:
[tex]\displaystyle{ AB= \sqrt{(-6+3)^2+(-1+4)^2}=\sqrt{9+9}=\sqrt{2\cdot9}=3\sqrt{2}[/tex].
[tex]\displaystyle{ AB= \sqrt{(6+3)^2+(5+4)^2}=\sqrt{81+81}=\sqrt{2\cdot81}=9\sqrt{2}[/tex].
[tex]\displaystyle{ AB= \sqrt{(6-3)^2+(5-8)^2}=\sqrt{9+9}=3\sqrt{2}[/tex].
[tex]\displaystyle{ AB= \sqrt{(-6-3)^2+(-1-8)^2}=\sqrt{81+81}=9\sqrt{2}[/tex].
Thus, the perimeter of the polygon is
[tex]9\sqrt{2}+9\sqrt{2}+3\sqrt{2}+3\sqrt{2}=24\sqrt{2}[/tex] (units).
Answer: [tex]24\sqrt{2}[/tex] units
