Compute the perimeter of the triangle determined by \( A(-2,-3), B(2,-6) \), and \( C(5,-5) \). \( A=\square \) unit \( ^{2} \)

To find the perimeter of the triangle defined by the points \( A(-2,-3) \), \( B(2,-6) \), and \( C(5,-5) \), we can calculate the lengths of each side using the distance formula \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). 1. Length of side \( AB \): \[ AB = \sqrt{(2 - (-2))^2 + (-6 - (-3))^2} = \sqrt{(2 + 2)^2 + (-6 + 3)^2} = \sqrt{(4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \] 2. Length of side \( BC \): \[ BC = \sqrt{(5 - 2)^2 + (-5 - (-6))^2} = \sqrt{(5 - 2)^2 + (-5 + 6)^2} = \sqrt{(3)^2 + (1)^2} = \sqrt{9 + 1} = \sqrt{10}. \] 3. Length of side \( CA \): \[ CA = \sqrt{(5 - (-2))^2 + (-5 - (-3))^2} = \sqrt{(5 + 2)^2 + (-5 + 3)^2} = \sqrt{(7)^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53}. \] Now, we sum the lengths of the sides to get the perimeter \( P \): \[ P = AB + BC + CA = 5 + \sqrt{10} + \sqrt{53}. \] Thus, the perimeter of the triangle is \( P = 5 + \sqrt{10} + \sqrt{53} \).