Suppose a parallelogram is drawn on a coordinate plane. Using vertices with the coordinates (2,1)%0D%0A(%0D%0A2%0D%0A,%0D%0A1%0D%0A)%0D%0A, (2,3)%0D%0A(%0D%0A2%0D%0A,%0D%0A3%0D%0A)%0D%0A, (4,4)%0D%0A(%0D%0A4%0D%0A,%0D%0A4%0D%0A)%0D%0A, and (4,2)%0D%0A(%0D%0A4%0D%0A,%0D%0A2%0D%0A)%0D%0A, compute the perimeter. Round your answer to the nearest tenth.(1 point)%0D%0AThe perimeter is approximately units.

Answer 1

To compute the perimeter of the parallelogram with the given vertices \((2, 1)\), \((2, 3)\), \((4, 4)\), and \((4, 2)\), we'll first calculate the lengths of its sides using the distance formula, which is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}. \]

We will calculate the distances between adjacent vertices of the parallelogram:

1. Distance between \((2, 1)\) and \((2, 3)\): \[ d_1 = \sqrt{(2 - 2)^2 + (3 - 1)^2} = \sqrt{0 + 2^2} = \sqrt{4} = 2. \]

2. Distance between \((2, 3)\) and \((4, 4)\): \[ d_2 = \sqrt{(4 - 2)^2 + (4 - 3)^2} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236. \]

3. Distance between \((4, 4)\) and \((4, 2)\): \[ d_3 = \sqrt{(4 - 4)^2 + (2 - 4)^2} = \sqrt{0 + (-2)^2} = \sqrt{4} = 2. \]

4. Distance between \((4, 2)\) and \((2, 1)\): \[ d_4 = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \approx 2.236. \]

Calculating the Perimeter

The perimeter \(P\) of the parallelogram is the sum of the lengths of all its sides:

\[ P = d_1 + d_2 + d_3 + d_4 = 2 + \sqrt{5} + 2 + \sqrt{5}. \]

Since \(d_2\) and \(d_4\) are both equal to \( \sqrt{5} \):

\[ P = 2 + 2 + 2\sqrt{5} = 4 + 2\sqrt{5}. \]

Now substituting the approximate value of \(\sqrt{5} \approx 2.236\):

\[ P \approx 4 + 2 \times 2.236 \approx 4 + 4.472 \approx 8.472. \]

Rounding to the Nearest Tenth

Thus, rounding \(8.472\) to the nearest tenth, we get:

\[ \text{Perimeter} \approx 8.5. \]

Therefore, the perimeter of the parallelogram is approximately \(8.5\) units.