To find the perimeter of the polygon with vertices [tex]\( G(2, 4) \)[/tex], [tex]\( H(2, -3) \)[/tex], [tex]\( J(-2, -3) \)[/tex], and [tex]\( K(-2, 4) \)[/tex], we need to calculate the distance between each pair of consecutive vertices, then sum these distances.
1. Calculate the distance between vertices [tex]\( G(2, 4) \)[/tex] and [tex]\( H(2, -3) \)[/tex] (denoted as [tex]\( GH \)[/tex]):
[tex]\[
GH = \sqrt{(2-2)^2 + (-3-4)^2} = \sqrt{0^2 + (-7)^2} = \sqrt{49} = 7
\][/tex]
2. Calculate the distance between vertices [tex]\( H(2, -3) \)[/tex] and [tex]\( J(-2, -3) \)[/tex] (denoted as [tex]\( HJ \)[/tex]):
[tex]\[
HJ = \sqrt{(-2-2)^2 + (-3-(-3))^2} = \sqrt{(-4)^2 + 0^2} = \sqrt{16} = 4
\][/tex]
3. Calculate the distance between vertices [tex]\( J(-2, -3) \)[/tex] and [tex]\( K(-2, 4) \)[/tex] (denoted as [tex]\( JK \)[/tex]):
[tex]\[
JK = \sqrt{(-2-(-2))^2 + (4-(-3))^2} = \sqrt{0^2 + (7)^2} = \sqrt{49} = 7
\][/tex]
4. Calculate the distance between vertices [tex]\( K(-2, 4) \)[/tex] and [tex]\( G(2, 4) \)[/tex] (denoted as [tex]\( KG \)[/tex]):
[tex]\[
KG = \sqrt{(2-(-2))^2 + (4-4)^2} = \sqrt{(4)^2 + 0^2} = \sqrt{16} = 4
\][/tex]
5. Sum these distances to find the perimeter:
[tex]\[
\text{Perimeter} = GH + HJ + JK + KG = 7 + 4 + 7 + 4 = 22
\][/tex]
Therefore, the perimeter of the polygon is [tex]\( \boxed{22} \)[/tex] units.
