To compute the perimeter of the triangle given by the coordinates (3, 4), (1, 1), and (5, 1), we will first calculate the lengths of the sides of the triangle using the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Let's find the lengths of each side of the triangle formed by the three points.
- Distance between (3, 4) and (1, 1):
\[ d_1 = \sqrt{(1 - 3)^2 + (1 - 4)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \text{ units} \]
- Distance between (1, 1) and (5, 1):
\[ d_2 = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4 \text{ units} \]
- Distance between (5, 1) and (3, 4):
\[ d_3 = \sqrt{(3 - 5)^2 + (4 - 1)^2} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.61 \text{ units} \]
Now we can calculate the perimeter of the triangle by adding the lengths of the sides:
\[ \text{Perimeter} = d_1 + d_2 + d_3 = \sqrt{13} + 4 + \sqrt{13} \approx 3.61 + 4 + 3.61 = 11.22 \text{ units} \]
Final Answer:
The perimeter of the triangle is approximately \( 11.22 \) units.