Use the coordinates to compute the perimeter of a paralleleogram whose vertices are located at the ordered pairs (0, 0), (1, 2), (2, 1) and (1, -1).

A. 7.2 units
B. 4 units

1 answer

To calculate the perimeter of the parallelogram with given vertices, we can follow these steps:

  1. Identify the vertices:

    • A = (0, 0)
    • B = (1, 2)
    • C = (2, 1)
    • D = (1, -1)
  2. Calculate the lengths of the sides: We will calculate the lengths of AB, BC, CD, and DA.

    • Length of AB: \[ AB = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

    • Length of BC: \[ BC = \sqrt{(2-1)^2 + (1-2)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

    • Length of CD: \[ CD = \sqrt{(1-2)^2 + (-1-1)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

    • Length of DA: \[ DA = \sqrt{(1-0)^2 + (-1-0)^2} = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

  3. Sum the lengths to find the perimeter: Since opposite sides of a parallelogram are equal, we can calculate the perimeter as follows: \[ \text{Perimeter} = 2(AB + BC) \] Plugging in the values we calculated: \[ \text{Perimeter} = 2(\sqrt{5} + \sqrt{2}) \]

  4. Numerical approximation: We can also calculate the approximate values of \(\sqrt{5}\) and \(\sqrt{2}\): \[ \sqrt{5} \approx 2.236, \quad \sqrt{2} \approx 1.414 \] Thus, \[ \text{Perimeter} \approx 2(2.236 + 1.414) \approx 2(3.650) \approx 7.3 \]

Since the values don't exactly match the options provided but are close to 7.2, the best estimate would be:

A. 7.2 units