Find the perimeter of the triangle with these vertices. \[ (1,5),(1,-4),(-4,-4) \] Give an exact answer (not a decimal approximation). Simplify your answer as much as possible.

To find the perimeter of the triangle with vertices \( (1,5) \), \( (1,-4) \), and \( (-4,-4) \), we first calculate the lengths of each side using the distance formula: The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] 1. **Length of side between points \( (1,5) \) and \( (1,-4) \)**: \[ d_1 = \sqrt{(1 - 1)^2 + (-4 - 5)^2} = \sqrt{0 + (-9)^2} = \sqrt{81} = 9 \] 2. **Length of side between points \( (1,-4) \) and \( (-4,-4) \)**: \[ d_2 = \sqrt{(-4 - 1)^2 + (-4 - (-4))^2} = \sqrt{(-5)^2 + 0} = \sqrt{25} = 5 \] 3. **Length of side between points \( (-4,-4) \) and \( (1,5) \)**: \[ d_3 = \sqrt{(1 - (-4))^2 + (5 - (-4))^2} = \sqrt{(1 + 4)^2 + (5 + 4)^2} = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106} \] Now, we can find the perimeter \( P \) of the triangle by adding all these lengths together: \[ P = d_1 + d_2 + d_3 = 9 + 5 + \sqrt{106} \] Thus, the exact perimeter of the triangle is: \[ \boxed{14 + \sqrt{106}} \]