To find the perimeter of a parallelogram, we first need to determine the lengths of its sides. The vertices of the parallelogram given are:
- \( A(2, 1) \)
- \( B(2, 3) \)
- \( C(4, 4) \)
- \( D(4, 2) \)
We can calculate the lengths of each side by using the distance formula between the vertices. The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculating Lengths of the Sides:
-
Length of side AB: \[ AB = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
-
Length of side BC: \[ BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
-
Length of side CD: \[ CD = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + 4} = \sqrt{4} = 2 \]
-
Length of side DA: \[ DA = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
Perimeter of the Parallelogram:
The perimeter \( P \) of a parallelogram can be calculated as: \[ P = 2(AB + BC) \] Using our calculations: \[ P = 2(2 + \sqrt{5}) \]
Approximating \(\sqrt{5}\):
The approximate value of \(\sqrt{5}\) is about \(2.236\).
Now substituting back into the perimeter equation: \[ P = 2(2 + 2.236) = 2(4.236) = 8.472 \]
Rounding to the Nearest Tenth:
\[ P \approx 8.5 \]
Thus, the perimeter of the parallelogram is approximately \( \boxed{8.5} \).