To compute the perimeter of the triangle formed by the coordinates \((3, 4)\), \((1, 1)\), and \((5, 1)\), we first need to calculate the lengths of each side 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 denote the points as follows:
- Point A: \((3, 4)\)
- Point B: \((1, 1)\)
- Point C: \((5, 1)\)
Calculating Distances
-
Distance AB: \[ AB = \sqrt{(1 - 3)^2 + (1 - 4)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Distance BC: \[ BC = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4 \]
-
Distance CA: \[ CA = \sqrt{(5 - 3)^2 + (1 - 4)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
Calculating the Perimeter
Now, we can calculate the perimeter \(P\) of the triangle by summing the lengths of the sides:
\[ P = AB + BC + CA = \sqrt{13} + 4 + \sqrt{13} = 2\sqrt{13} + 4 \]
Now, let's approximate \(\sqrt{13}\): \[ \sqrt{13} \approx 3.605 \] So, \[ P \approx 2(3.605) + 4 = 7.21 + 4 \approx 11.21 \]
Thus, the perimeter of the triangle is approximately 11.2 units.
Therefore, the correct response is: 11.2 units.