Find the perimeter of the polygon with the vertices [tex]\(G(2,4), H(2,-3), J(-2,-3),\)[/tex] and [tex]\(K(-2,4)\)[/tex].

The perimeter is [tex]\(\square\)[/tex] units.



Answer :

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.