ind the perimeter of the polygon with the vertices \( G(2,4), H(2,-3), J(-2,-3) \), and \( K(-2,4) \)

To find the perimeter of the polygon with the vertices \( G(2,4) \), \( H(2,-3) \), \( J(-2,-3) \), and \( K(-2,4) \), we first need to calculate the lengths of the sides of the polygon by finding the distances between consecutive vertices. 1. **Distance \( GH \)**: \[ GH = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - 2)^2 + (-3 - 4)^2} = \sqrt{0 + (-7)^2} = \sqrt{49} = 7 \] 2. **Distance \( HJ \)**: \[ HJ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-2 - 2)^2 + (-3 - (-3))^2} = \sqrt{(-4)^2 + 0} = \sqrt{16} = 4 \] 3. **Distance \( JK \)**: \[ JK = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-2 - (-2))^2 + (4 - (-3))^2} = \sqrt{0 + (4 + 3)^2} = \sqrt{7^2} = 7 \] 4. **Distance \( KG \)**: \[ KG = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - (-2))^2 + (4 - 4)^2} = \sqrt{(2 + 2)^2 + 0} = \sqrt{4^2} = 4 \] Now, we can sum the lengths of all the sides to find the perimeter \( P \): \[ P = GH + HJ + JK + KG = 7 + 4 + 7 + 4 = 22 \] Thus, the perimeter of the polygon is \( \boxed{22} \).