To find the perimeter of a triangle, we need to calculate the lengths of its sides using the distance formula. The formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 1: Calculate the distances between each pair of vertices.

Let the vertices be $A(-1, 5)$, $B(-1, 1)$, and $C(-4, -1)$.

1. Distance between A and B:
   $AB = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{(-1 - (-1))^2 + (1 - 5)^2} = \sqrt{(0)^2 + (-4)^2} = \sqrt{16} = 4$

2. Distance between B and C:
   $BC = \sqrt{(x_C - x_B)^2 + (y_C - y_B)^2} = \sqrt{(-4 - (-1))^2 + (-1 - 1)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}$

3. Distance between A and C:
   $AC = \sqrt{(x_C - x_A)^2 + (y_C - y_A)^2} = \sqrt{(-4 - (-1))^2 + (-1 - 5)^2} = \sqrt{(-3)^2 + (-6)^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$

Step 2: Add up the distances to find the perimeter.

Now, the perimeter $P$ is the sum of the lengths of the three sides:

$P = AB + BC + AC = 4 + \sqrt{13} + 3\sqrt{5}$

Thus, the exact perimeter of the triangle is:

$P = 4 + \sqrt{13} + 3\sqrt{5}$

Would you like further details or explanations on any step? Here are some related questions for deeper exploration:

1. How does the distance formula relate to the Pythagorean theorem?
2. What would happen if the vertices were in a straight line (collinear)?
3. How do you calculate the area of the triangle from the vertices?
4. What is the relationship between the lengths of the sides and the type of triangle?
5. How would the perimeter change if one of the vertices were moved along the x-axis?

Tip: Always ensure to calculate exact values for geometric measurements to avoid rounding errors when working with exact formulas.