To calculate the perimeter of the parallelogram formed by the vertices \((0,0)\), \((1,2)\), \((2,1)\), and \((1,-1)\), we first need to find the lengths of the sides.
Step 1: Calculate the lengths of the sides.
We can calculate the lengths of the sides using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Length of side 1: Between (0,0) and (1,2)
\[ d_1 = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]
Length of side 2: Between (1,2) and (2,1)
\[ d_2 = \sqrt{(2 - 1)^2 + (1 - 2)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]
The opposite sides of the parallelogram will have the same lengths, hence:
- The length of the side between (0,0) and (1,2) is the same as the length of the side between (1,-1) and (2,1).
- The length of the side between (1,2) and (2,1) is the same as the length of the side between (0,0) and (1,-1).
Step 2: Calculate the perimeter.
The perimeter \(P\) of a parallelogram is given by:
\[ P = 2 \times (d_1 + d_2) \]
Substitute in the values we found:
\[ P = 2 \times (\sqrt{5} + \sqrt{2}) \]
Now we can approximate the numerical values:
- \(\sqrt{5} \approx 2.236\)
- \(\sqrt{2} \approx 1.414\)
So:
\[ P \approx 2 \times (2.236 + 1.414) \approx 2 \times 3.650 \approx 7.3 \text{ (approximately)} \]
Final Answer:
From the options given, none match this exactly. However, rounding might lead closely to:
7.2 units
Therefore, while exact calculation suggests around \(7.3\), the closest matching option is:
7.2 units