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], follow these steps to calculate the distance between each pair of consecutive vertices:
1. Calculate the distance between [tex]\((-2,1)\)[/tex] and [tex]\((-2,7)\)[/tex]:
[tex]\[
d_1 = \sqrt{( -2 - (-2) )^2 + ( 7 - 1 )^2} = \sqrt{0 + 6^2} = \sqrt{36} = 6
\][/tex]
2. Calculate the distance between [tex]\((-2,7)\)[/tex] and [tex]\((1,11)\)[/tex]:
[tex]\[
d_2 = \sqrt{( 1 - (-2) )^2 + ( 11 - 7 )^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
3. Calculate the distance between [tex]\((1,11)\)[/tex] and [tex]\((4,7)\)[/tex]:
[tex]\[
d_3 = \sqrt{( 4 - 1 )^2 + ( 7 - 11 )^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\][/tex]
4. Calculate the distance between [tex]\((4,7)\)[/tex] and [tex]\((4,1)\)[/tex]:
[tex]\[
d_4 = \sqrt{( 4 - 4 )^2 + ( 1 - 7 )^2} = \sqrt{0 + (-6)^2} = \sqrt{36} = 6
\][/tex]
5. Calculate the distance between [tex]\((4,1)\)[/tex] and [tex]\((-2,1)\)[/tex]:
[tex]\[
d_5 = \sqrt{( -2 - 4 )^2 + ( 1 - 1 )^2} = \sqrt{(-6)^2 + 0} = \sqrt{36} = 6
\][/tex]
Now, sum the distances to find the perimeter of the polygon:
[tex]\[
\text{Perimeter} = d_1 + d_2 + d_3 + d_4 + d_5 = 6 + 5 + 5 + 6 + 6 = 28
\][/tex]
Thus, the perimeter of the polygon is:
[tex]\[
\boxed{28.0}
\][/tex]
