Answer: \(4 + \sqrt{13} + 3\sqrt{5}\) units
Explanation:
To find the perimeter of a triangle with vertices (-1, 5), (-1, 1), and (-4, -1), you can use the distance formula to calculate the lengths of each side of the triangle. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Let's calculate the distances for each side:
1. Distance between (-1, 5) and (-1, 1):
\[
\text{Distance}_1 = \sqrt{(-1 - (-1))^2 + (1 - 5)^2} = \sqrt{0^2 + (-4)^2} = \sqrt{16} = 4
\]
2. Distance between (-1, 1) and (-4, -1):
\[
\text{Distance}_2 = \sqrt{(-4 - (-1))^2 + (-1 - 1)^2} = \sqrt{(-3)^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]
3. Distance between (-4, -1) and (-1, 5):
\[
\text{Distance}_3 = \sqrt{(-1 - (-4))^2 + (5 - (-1))^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}
\]
Now, you can find the perimeter by adding up the lengths of all three sides:
\[
\text{Perimeter} = \text{Distance}_1 + \text{Distance}_2 + \text{Distance}_3 = 4 + \sqrt{13} + 3\sqrt{5}
\]
So, the exact perimeter of the triangle is \(4 + \sqrt{13} + 3\sqrt{5}\) units.