To find the perimeter of the given polygon with vertices at [tex]\((-2,1)\)[/tex], [tex]\((-2,7)\)[/tex], [tex]\((1,11)\)[/tex], [tex]\((4,7)\)[/tex], and [tex]\((4,1)\)[/tex], we will follow these steps:
1. Calculate the lengths of each side using the distance formula:
The distance formula to find the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{{(x_2 - x_1)^2 + (y_2 - y_1)^2}}
\][/tex]
Side 1: Between [tex]\((-2,1)\)[/tex] and [tex]\((-2,7)\)[/tex]
[tex]\[
\text{distance} = \sqrt{{(-2 - (-2))^2 + (7 - 1)^2}} = \sqrt{{0 + 6^2}} = \sqrt{{36}} = 6
\][/tex]
Side 2: Between [tex]\((-2,7)\)[/tex] and [tex]\((1,11)\)[/tex]
[tex]\[
\text{distance} = \sqrt{{(1 - (-2))^2 + (11 - 7)^2}} = \sqrt{{3^2 + 4^2}} = \sqrt{{9 + 16}} = \sqrt{{25}} = 5
\][/tex]
Side 3: Between [tex]\((1,11)\)[/tex] and [tex]\((4,7)\)[/tex]
[tex]\[
\text{distance} = \sqrt{{(4 - 1)^2 + (7 - 11)^2}} = \sqrt{{3^2 + (-4)^2}} = \sqrt{{9 + 16}} = \sqrt{{25}} = 5
\][/tex]
Side 4: Between [tex]\((4,7)\)[/tex] and [tex]\((4,1)\)[/tex]
[tex]\[
\text{distance} = \sqrt{{(4 - 4)^2 + (1 - 7)^2}} = \sqrt{{0 + (-6)^2}} = \sqrt{{36}} = 6
\][/tex]
Side 5: Between [tex]\((4,1)\)[/tex] and [tex]\((-2,1)\)[/tex]
[tex]\[
\text{distance} = \sqrt{{(4 - (-2))^2 + (1 - 1)^2}} = \sqrt{{6^2 + 0}} = \sqrt{{36}} = 6
\][/tex]
2. Sum the lengths of all the sides to find the perimeter:
[tex]\[
\text{Perimeter} = 6 + 5 + 5 + 6 + 6 = 28
\][/tex]
So, the perimeter of the polygon is [tex]\(28\)[/tex] units.
Thus, the answer is:
[tex]\[
\boxed{28}
\][/tex]
