To find the perimeter of the parallelogram with the given vertices \((2,1)\), \((2,3)\), \((4,4)\), and \((4,2)\), we first need to calculate the lengths of the sides.
Step 1: Identify the pairs of vertices:
We can pair the vertices as follows:
- Vertices A(2, 1) and B(2, 3)
- Vertices C(4, 4) and D(4, 2)
Step 2: Calculate the lengths of the sides.
The lengths of the sides can be calculated using the distance formula for points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Length of AB:
\[ AB = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2 \]
Length of BC:
\[ BC = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]
The lengths of the sides of a parallelogram opposite sides are equal. So:
- \(AB\) and \(DC\) are both of length \(2\).
- \(BC\) and \(AD\) are both of length \(\sqrt{5}\).
Step 3: Calculate the perimeter.
The perimeter \(P\) of a parallelogram is given by: \[ P = 2(AB + BC) = 2(2 + \sqrt{5}) \]
Calculating \(\sqrt{5}\): \[ \sqrt{5} \approx 2.236 \]
Now substituting this into the perimeter: \[ P \approx 2(2 + 2.236) = 2(4.236) \approx 8.472 \]
Step 4: Round to the nearest tenth.
Rounding \(8.472\) gives: \[ P \approx 8.5 \]
Thus, the perimeter of the parallelogram is approximately 8.5 units.