Find the perimeter of the parallelogram with these vertices. \( (3,6),(-5,6),(6,-1),(-2,-1) \) Give an exact answer (not a decimal approximation). Simplify your answer as much as possible.

To find the perimeter of the parallelogram, we first need to determine the lengths of the sides by calculating the distance between two consecutive vertices. We start with the vertices given: \( A(3,6), B(-5,6), C(6,-1), D(-2,-1) \). 1. **Finding the length of side AB**: \[ AB = \sqrt{((-5) - 3)^2 + (6 - 6)^2} = \sqrt{(-8)^2 + 0^2} = \sqrt{64} = 8 \] 2. **Finding the length of side BC**: \[ BC = \sqrt{(6 - (-5))^2 + ((-1) - 6)^2} = \sqrt{(6 + 5)^2 + (-1 - 6)^2} = \sqrt{11^2 + (-7)^2} = \sqrt{121 + 49} = \sqrt{170} \] Since opposite sides of a parallelogram are equal, both sides \( AB \) and \( CD \) will also have length \( 8 \), and both sides \( BC \) and \( DA \) will also have length \( \sqrt{170} \). 3. **Calculating the perimeter**: The perimeter \( P \) of the parallelogram is given by: \[ P = 2 \times (AB + BC) = 2 \times (8 + \sqrt{170}) = 16 + 2\sqrt{170} \] Therefore, the perimeter of the parallelogram is: \[ \boxed{16 + 2\sqrt{170}} \]