To find the perimeter of the parallelogram with the given vertices (0,0), (2,3), (5,2), and (3,-1), we first need to calculate the lengths of the sides. The perimeter of a parallelogram is given by the sum of the lengths of all its sides. Since opposite sides are equal in length, we can calculate the lengths of just two adjacent sides and multiply by 2.
Step 1: Calculate the lengths of two sides
Length of side AB
Let \( A(0,0) \) and \( B(2,3) \). The length AB can be calculated using the distance formula: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates: \[ AB = \sqrt{(2 - 0)^2 + (3 - 0)^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]
Length of side BC
Let \( B(2,3) \) and \( C(5,2) \). Using the distance formula again: \[ BC = \sqrt{(5 - 2)^2 + (2 - 3)^2} = \sqrt{3^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]
Step 2: Calculate the perimeter
The perimeter \( P \) is given by: \[ P = 2(AB + BC) = 2(\sqrt{13} + \sqrt{10}) \]
Step 3: Calculate numerical values
Now we will calculate the square roots and the perimeter: \[ \sqrt{13} \approx 3.6056 \] \[ \sqrt{10} \approx 3.1623 \]
Now plug these into the perimeter formula: \[ P \approx 2(3.6056 + 3.1623) \approx 2(6.7679) \approx 13.5358 \]
Step 4: Round to the nearest tenth
Thus, the perimeter of the parallelogram is approximately: \[ \boxed{13.5} \text{ units.} \]
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